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6651 tp.indd 1 9/28/07 4:50:42 PM
T Trigonometric Identity Proofs
(TIPs)
Intellectually challenging games
This page intentionally left blankThis page intentionally left blank
N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I
World Scientific
Y E O AdrianM.A., Ph.D. (Cambridge University)
Honorary Fellow, Christ’s College, Cambridge University
6651 tp.indd 2 9/28/07 4:50:44 PM
An encyclopedia ofTrigonometric Identity Proofs
Intellectually challenging games
(TIPs)
British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.
For photocopying of material in this volume, please pay a copying fee through the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission tophotocopy is not required from the publisher.
ISBN-13 978-981-277-618-1ISBN-10 981-277-618-4ISBN-13 978-981-277-619-8 (pbk)ISBN-10 981-277-619-2 (pbk)
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,electronic or mechanical, including photocopying, recording or any information storage and retrievalsystem now known or to be invented, without written permission from the Publisher.
Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd.
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Printed in Singapore.
TRIG OR TREATAn Encyclopedia of Trigonometric Identity Proofs (TIPs) with IntellectuallyChallenging Games
SC - Trig or Treat.pmd 9/17/2007, 10:43 AM1
September 12, 2007 19:16 Book:- Trig or Treat (9in x 6in) 01˙dedi
dedicated to the memoryof my mother
tan peck hiah
tan A+ tan B+ tan C ≡ tan A tan B tan C
(A+B+C = 180◦)
tan A+ tan(A+120◦)+ tan(A+240◦) = 3 tan(3A+360◦)
tan−1
(
11
)
= tan−1
(
12
)
+ tan−1
(
13
)
1π
=122 tan
π22 +
123 tan
π23 +
124 tan
π24 +
125 tan
π25 + · · ·
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September 17, 2007 19:18 Book:- Trig or Treat (9in x 6in) 02˙contents
Contents
Preface ix
Introduction xiii
Trig — Level One 1
The Basics of Trigonometry 1Pythagoras’ Theorem 11
Trig — Level Two 15
Compound Angles, Double Angles and Half Angles
Trig — Level Three 25
Angles in a TriangleSum and Difference of sin and cos
Practical Trig 33
Numerical Values of Special Angles —Graphs of sin, cos and tan for 0◦–360◦
Appendix 45
The Concordance of Trigonometric Identities 47The Encyclopedia of Trigonometric Games or
Trigonometric Identity Proofs (TIPs) 71
vii
September 17, 2007 19:18 Book:- Trig or Treat (9in x 6in) 02˙contents
viii Trig or Treat
Level-One-Games: Easy Proofs 73
Level-Two-Games: Less-Easy Proofs 251
Level-Three-Games: Not-So-Easy Proofs 349
Addenda 393
September 17, 2007 22:30 Book:- Trig or Treat (9in x 6in) 03˙preface
Preface
Many students find “Trigonometry” to be a difficult topic in a diffi-cult subject “Math”. Yet most students have no difficulty with com-puter games, and enjoy playing them even though many of thesegames have lots of pieces to manipulate, and are subjected to com-plex rules. For example, a simple game like “Tetris” has seven differ-ent pieces; and the player has to orientate and manipulate each piecein turn, as it falls. The objective is to construct a solid wall with all therandom pieces — and to do it, racing against the clock.
“Trigonometry”, or “Trig” for short, can be thought of as an intellec-tual equivalent of “Tetris”. There are six main pieces to manipulate. Threeof them, sine, cosine, and tangent, are most important — that is why thisbranch of Math is called Trigonometry; the “tri” refers to three functions,three angles and three sides of a triangle. And there is only one simple rule— Logic. Trig can be thought of as a game that involves the logical manip-ulation of various trig pieces to achieve different identities and equations,and to solve numerical problems.
Trig can also be viewed as a non-numerical equivalent of the numbergame “Sudoku”. The logic and the arrangement of the digits 1 to 9, is nowapplied to the six trig pieces — sine, cosine, tangent, cosecant, secant andcotangent.
Tetris and Sudoku are both simple games that give lots of fun andpleasure. Trig is also a simple game, but with a vital difference —knowledge of it has invaluable applications in Math, surveying, building,
ix
September 17, 2007 22:30 Book:- Trig or Treat (9in x 6in) 03˙preface
x Trig or Treat
navigation, astronomy, and other branches of science, engineering, andtechnology.
Adults, children and students can play Sudoku and Tetris for hours onend. So they should have little difficulty playing “Trig”, if they derivesimilar fun and pleasure from it.
Albert Einstein said:
“Everything should be made as simple as possible,But not simpler”.
This book seeks to make Trig as simple as possible, by treating it as agame — albeit, an intellectual game — as interesting and stimulating asTetris and Sudoku. Mastering of Trig will not only give mental and intel-lectual satisfaction and pleasure, but it will also lead to beneficial results inone’s future career and life.
This book is the third math book∗ that I have written for my two grand-children, Kathryn and Rebecca, ages 5 and 7, respectively. The challengethat I set for myself here is to explain Trig so simply that my seven-year-old granddaughter, Rebecca, can understand it. Indeed she has been able todo some of the “Level-One-Games”. My hope is that in the coming yearsboth Kathryn and Rebecca would be able to play the “Level-Two-Games”and the “Level-Three-Games” in this book also.
I thank Lim Sook Cheng and her excellent team at World ScientificPublishing for the production of this book; and Zee Jiak Gek for her metic-ulous reading and critique of all the details in the manuscript.
∗ The other two books are “ The Pleasures of Pi,e and Other Interesting Numbers”,and “Are You the King or Are You the Joker?”.
September 17, 2007 22:30 Book:- Trig or Treat (9in x 6in) 03˙preface
xi
September 17, 2007 22:30 Book:- Trig or Treat (9in x 6in) 03˙preface
xii
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 04˙introduction
Introduction
The approach taken in this book is to treat Trig as a game. Beginning withonly the definition of sine, the superstar of Trig, the book introduces thereader slowly to the basics of Trig. Then, by applying simple logic, the twoco-stars, cosine and tangent, are introduced.
Thereafter three supporting starlets, named the reciprocals — cosecant,secant and cotangent — are added. With these six pieces, the applicationof simple logic, arithmetic and algebra will give countless Trig equationscalled identities. Played like jigsaw puzzles, Tetris and Sudoku, movingthe Trig pieces around to give different identities can be a lot of fun.
As with other games and puzzles, practice can lead to greater skill andmental agility. About 300 games (proofs) are provided in this book to givefun (and confidence) to readers who want to try their hands (and worktheir brains) on these intellectual games. The numerous games are broadlygrouped into three overlapping levels — Level-One-Games (Easy Proofs),Level-Two-Games, (Less-Easy Proofs) and Level-Three-Games (Not-so-Easy Proofs).
For the first time ever, a “Concordance of Trigonometric Identities”has been created. Trigonometric identities are given a 6-digit code, whichenables readers (and students) to have easy reference to the identity to beproved, and to locate rapidly the proof in the Encyclopedia of Trigonomet-ric Identity Proofs (TIPs) in the Appendix.
Readers are welcome to look at the identities in the Concordance first,and try their hand at proving any of the identities, prior to looking at the
xiii
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 04˙introduction
xiv Trig or Treat
detailed proofs in the Encyclopedia. (Some identities which may appearsimple, may be difficult to prove; conversely, some complex-looking iden-tities may turn out to be relatively easy!)
The games provide the challenge to readers to match their skills, andprogress up the ladder of increasing intellectual agility. If you are reallygood in Trig, then the speed of proving the identities is the speed withwhich you write out the proofs, i.e. your brain works faster than your brawn(hand).
Have fun with Trig!
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1
Trig — Level One
The Basics of Trigonometry
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1
2 Trig or Treat
90°
90°
0° 60°
45° 30°
Acute Angles
120°
150° 135°
180°
90°
180°
Obtuse Angles
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1
The Basics of Trigonometry 3
Measuring Angles
“The sun rises in the east, and sets in the west”. Similarly, the measur-ing of angles begins “in the east” (0◦), goes counterclockwise, up into theoverhead sky at noon (90◦) and sets in the west (180◦).
People in many ancient civilisations (including the Babylonians,Mesopotamians and the Egyptians) used a numbering system based on 60called the sexagesimal system. This resulted in the convention of 360◦
(60◦ × 6) for the angle round a point. This convention for measuringangles continues to the present day, despite the widespread use of themetric system based on decimals (10’s). Another sexagesimal legacy fromthe past is the use of 60 seconds in a minute, and 60 minutes in an hour.
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1
4 Trig or Treat
Sine
A
HypotenuseOpposite side of angle A
90°
sine A =
sin A =
OppositeHypotenuse
OH
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1
The Basics of Trigonometry 5
Sine
Over the centuries, many civilisations used calculations based on right-angled triangles and the relationships of their sides for various purposes,including the building of monuments such as palaces, temples, and pyra-mids and other tombs for their rulers. Some of these mathematical tech-niques were also applicable to the study of the stars (astronomy) which ledto calender making.
The origins of Trig are lost in the mist of antiquity. One of the earliestrecorded reference to the concept of “the sine of an angle” — “jya” — wasfound in a sixth century Indian math book. This word was later translatedinto “jiba” or “jaib” in Arabic. A further translation into Latin convertedthe word into “sinus”, meaning a bay or curve, the same meaning as “jaib”.This was further simplified in the 17th century into English — “sine”, andabbreviated as “sin” (but always pronounced as “SINE”and not “SIN”.)
Sine is simply the name of a specific ratio:
sine of an angle (A) =length of the opposite side of angle (A)
length of the hypotenuse
This definition is often abbreviated to
sin A =OH
You cannot do Trig if you cannot remember the definition of sin! Thereare many simple ways of remembering. How about:
1. O/H lang SINE?∗
2. O/H , it’s so SINple?
Can you create your own mnemonics?
∗Sounds like “Auld Lang Syne”, the universally popular song sung at the strokeof midnight on New Year’s Day.
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1
6 Trig or Treat
Cosine
Hypotenuse
Adjacent side of angle A
(90−A)°
Adjacent
Hypotenuse
AH
cos A =
=
A 90°
Tangent
Adjacent side of angle A
Opposite side of angle A
Opposite
Adjacent
= OA
tan A =
sin Acos A
tan A =
A 90°
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1
The Basics of Trigonometry 7
Cosine and Tangent
The complementary angle to the angle A in a right-angled triangle is thethird angle, with the value of (90−A)◦, because the three angles of a tri-angle sum to a total of 180◦. The term “co-sine” was derived from thephrase “the sine of the complementary angle”
co-sine A = sine of complementary angle of A
= sine(90−A)◦
∴ cos A =length of the adjacent side of angle A
length of the hypotenuse
=AH
Tangent∗ is defined as the ratio of sin A/cos A
∴ tan A =sin Acos A
=
OHAH
=OA
∗This ratio (tangent) should be distinguished from the line which touches a circle,which is also called tangent in geometry.
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1
8 Trig or Treat
Reciprocals
cosec A =1
sin A
sec A =1
cos A
cot A =1
tan A
tan AA =sin AAcos AA
cot AA =cos AAsin AA
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1
The Basics of Trigonometry 9
Reciprocals
The superstar “sin” and its two co-stars (cos and tan) make up the three keyplayers in Trig. Their definitions and their relationships are essential for allproblems in Trig. Hence it is important that they be committed to memory.
Three more trig terms — the supporting cast — are also used. Theseare known as the reciprocals, and are best remembered as the reciprocalsof sin, cos and tan.
1sin A
= cosec A (cosecant)
1cos A
= sec A (secant)
1tan A
= cot A (cotangent)
These reciprocals are rarely used in applications in science, engineeringand technology. But for intellectual gymnastics (and in examinations!),these reciprocals are often used in equations and identities.
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1
10 Trig or Treat
Pythagoras’ Theorem
b
a2 + b2 = c2
a
c
The Famous “3-4-5 Triangle”
5 3
4
32 + 42 = 52
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1
The Basics of Trigonometry 11
Pythagoras’ Theorem∗
The most well-known theorem in Math, which practically every student haslearnt, is the Pythagoras’ Theorem, named after the Greek mathematicianPythagoras (∼580–500 BC).
This theorem states that in a right-angled triangle, the square of thehypotenuse (the longest side) is equal to the sum of the squares of thetwo other sides. This lengthy statement can be represented accurately inmathematical terms:
a2 +b2 = c2
where a and b are lengths of the two sides, and c is the length of thehypotenuse, the side facing the right-angle.
The most famous right-angled triangle is the “3-4-5 triangle”:
32 +42 = 52
(9+16 = 25)
A less famous sister is the “5-12-13 triangle” (52 + 122 = 132;25 + 144 = 169).
Recent research has shown that many civilisations, including the Baby-lonian, the Egyptian, the Chinese and the Indian civilisations, indepen-dently knew about the relationship between the squares of the three sidesof the right-angled triangle, in some cases, centuries before Pythagoras wasborn. (This illustrates a truism in Math, that often, your discoveries basedon your own efforts, may have been preceded by others. However this doesnot diminish in any way, the pleasure, excitement and sense of achievementthat you experienced — the so-called “eureka effect”. Indeed, it proves thatyou have a mathematical mind, capable of the same deep thoughts as theancient heroes of Math.)
∗A theorem is simply a mathematical statement whose validity has been provenby meticulous mathematical reasoning.
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1
12 Trig or Treat
Trig Equivalent of Pythagoras’ Theorem
b
a
c
A
A
cos A
1sin A
sin2 A + cos2 A = 1
cos2 A
sin2 A = 1 − cos2 A
= 1 − sin2 A
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1
The Basics of Trigonometry 13
Trig Equivalent of Pythagoras’ Theorem
One of the most important of Trig identities∗ is the trig equivalent of thePythagoras’ Theorem. The proof is simple:
sin A =ac
cos A =bc
sin2 A+ cos2 A =a2
c2 +b2
c2
=a2 +b2
c2
=c2
c2
(
by Pythagoras’ Theorem
a2 +b2 = c2
)
= 1
sin2AA+ cos2AA ≡ 1
A simpler visual proof can be obtained by using a special right-angledtriangle with a hypotenuse of unit length (1).
Then the length of the opposite side is now equal to sin A, and thelength of the adjacent side is equal to cos A (see figure opposite). Thenby Pythagoras’ Theorem:
sin2 A+ cos2 A ≡ 12
sin2 A+ cos2 A ≡ 1 .
This unity trig identity is the simplest and the most important of all trigidentities. It is also extremely useful in helping to solve trig problems.Whenever you see sin2 A or cos2 A, always consider the possibility of usingthis identity to simplify further.
∗An identity is a mathematical equation that is true for all values of the angle A. Itdoes not matter whether A = 30◦, 60◦, 90◦ etc, whether it is acute or obtuse, etc.The symbol (≡) is used to show that the two sides of an equation are identical.
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1
14 Trig or Treat
We can also derive two other identities:
dividingby sin2 A
sin2 A
sin2 A+
cos2 A
sin2 A≡
1
sin2 A
1+ cot2 A ≡ cosec2A
dividingby cos2 A
sin2 Acos2 A
+cos2 Acos2 A
≡
1cos2 A
tan2 A+1 ≡ sec2A.
After this simple introduction, you are now ready to play Level-One-Games (Easy Proofs), some of which seven-year-old Rebecca could play.
The general approach for playing the games (proving the identities)is to:
1. start with the more complex side of the identity (usually the left handside (LHS));
2. eyeball the key terms, and think in terms of sin and cos of the angle;3. engage in some mental gymnastics — rearranging and simplifying;4. whilst at all times, keeping the terms in the right hand side — the final
objective — in mind.
Like a guided missile, your logic and math manipulation of the LHSshould lead you to zoom in to the RHS.
Have Fun!
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 06˙level-2
Trig — Level Two
Compound Angles
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 06˙level-2
16 Trig or Treat
Sine of (the Sum of Two Angles)
sin(A+B) = sin A cos B+ cos A sin B
Substituting B = A,
sin(A+A) = sin A cos A+ cos A sin A
∴ sin 2A = 2 sin A cos A (Double Angle Formula)
Substituting A with (A/2),
sin 2
(
A2
)
= 2 sinA2
cosA2
∴ sin A = 2 sinA2
cosA2
(Half Angle Formula)
Sine of (the Difference of Two Angles)
Substituting B in the first equation by (−B),
sin (A+(−B)) = sin A cos(−B)+ cos A sin(−B)
∴ sin(A−B) = sin A cos B− cos A sin B
since cos(−B) = cos B
and sin(−B) = −sin B
See Proof in Addenda p. 395
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 06˙level-2
Compound Angles 17
Sine of the Sum and the Difference of Angles
This is the beginning of Level-Two-Games. The sin of compound angles(i.e. angles that are the sum or the difference of two other angles) can beexpressed in terms of trig functions of the single angles.
For example, sin of (A + B) can be expressed as a combination of thesum and products of the sin and cos of A and B separately. Similarly, sincewe know that sin(−A) = −sin A, and cos(−A) = cos A, we can derivesin(A−B). By judicious substitution (using B = A), sin(A + B) can bechanged to sin 2A and then into sin A (using A = 2(A/2)).
In earlier days, before calculators and computers were available, knowl-edge of the trig functions of compound angles was invaluable in practicalworkplace calculations. Knowing the basics for 0◦, 30◦, 45◦, 60◦ and 90◦,one could work out the values of trig functions of 15◦ and 22.5◦ and othersuch angles by way of these functions. In those days, Trig was both “puremath” and “applied math”, useful in many professions involving science,engineering and architecture.
Today, where the pressing of a few buttons on a calculator or computerwill give answers for all such calculations, Trig is largely “pure math” —a mental pursuit, an intellectual game.
But what a beautiful game it is, especially when you are immersed inthe proving of the vast number of identities!
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 06˙level-2
18 Trig or Treat
Cosine of (the Sum of Two Angles)
cos(A+B) = cos A cos B− sin A sin B
Substituting B = A,
cos 2A = cos2 A− sin2 A (Double Angle Formula)
= 2 cos2 A−1
= 1−2 sin2 A
Note the minussign on the RHS,even though thesign on the LHSis plus
Remember:-
s2 + c2 = 1
∴ −s2 = c2−1
and c2 = 1− s2
Substituting A with (A/2),
cos 2
(
A2
)
= cos2 A2− sin2 A
2
∴ cos A = cos2 A2− sin2 A
2(Half Angle Formula)
= 2 cos2 A2−1
= 1−2 sin2 A2
Cosine of (the Difference of Two Angles)
Substituting B in the first equation by (−B),
cos(A+(−B)) = cos A cos(−B)− sin A sin(−B)
∴ cos(A−B) = cos A cos B+ sin A sin B
since cos(−B) = cos B
and sin(−B) = −sin B
Note the plussign on the RHS,even though thesign on the LHSis minus
See Proof in Addenda p. 397
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 06˙level-2
Compound Angles 19
Cosine of the Sum and Difference of Angles
Cos functions require special attention as occassionally they act in a con-trarian, counter-intuitive manner — this arises largely from the fact thatcos(−A) = cos A.
This is the first function where such counter-intuitive behaviour of cosshows itself.
Although the LHS of the cos function is for the sum of two angles, theRHS shows a difference of the two products. Students are usually carelessand are not sufficiently sensitive to such minor (???) intricacies in Math.Unfortunately such minor (???) inattention can be very costly in examina-tions because they lead to wrong answers and major (???) losses in marks!
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 06˙level-2
20 Trig or Treat
Tangent of (the sum of Two Angles)
tan(A+B) =tan A+ tan B
1− tan A tan BNote the minussign in thedenominator onthe RHS, eventhough the LHShas a plus sign!
Substituting B = A,
tan 2A =2 tan A
1− tan2 A(Double Angle Forumla)
Substituting A with (A/2),
tan 2
(
A2
)
=2 tan
A2
1− tan2 A2
(1)
∴ tan A =2 tan
A2
1− tan2 A2
(Half Angle Formula) (2)
Tangent of (the Difference of Two Angles)
Substituting B in the first equation by (−B),
tan(A+(−B)) =tan A+ tan(−B)
1− tan A tan(−B)
∴ tan(A−B) =tan A− tan B
1+ tan A tan B
since tan(−B) = − tan B
Note the plus signin the denomina-tor on the RHS,even though theLHS has a minussign!
See Proof in Addenda p. 398
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 06˙level-2
Compound Angles 21
Tangent of the Sum and the Difference of Two Angles
The tan formulas for the sum and difference of two angles derive directlyfrom the sin and cos formulas. The double angle formula, tan(2A), andthe half angle formula, tan A, have proven to be extremely valuable inmany mathematical proofs, and have resulted in sophisticated methods forthe calculation of π to a large number of decimal places. (Would youbelieve that π has been calculated to 1.24 trillion decimal places — yes,1,240,000,000,000 decimals?)
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 06˙level-2
22 Trig or Treat
Summary of Trig Functions for Compound Angles
sin(A+B) = sin A cos B+ cos A sin B
sin 2A = 2 sin A cos A
sin A = 2 sinA2
cosA2
sin(A−B) = sin A cos B− cos A sin B
cos(A+B)= cos A cos B− sin A sin B
cos 2A = cos2 A− sin2 A
cos A = cos2 A2− sin2 A
2
cos(A−B)= cos A cos B+ sin A sin B
tan(A+B) =tan A+ tan B
1− tan A tan B
tan 2A =2 tan A
1− tan2 A
tan A =2 tan
A2
1− tan2 A2
tan(A−B) =tan A− tan B
1+ tan A tan B
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 06˙level-2
Compound Angles 23
The sin, cos and tan of compound angles, and their “double angle” and“half angle” formulas provide the basis for many of the Level-Two-Games.
Together with the most important identity:
sin2 A+ cos2 A ≡ 1 ,
the 12 identities on the opposite page, make up the total of the key Trigfunctions. Most other Trig identities can be derived from these “12 + 1”key identities.
The typical student (or Trig player) is expected to know these “12+1”key functions (very lucky if you know them instinctively, and very un-lucky if you don’t). With these “12+1” key functions, Level-Two-Games(Less-Easy Proofs) should prove to be easy also.
Within Level-Two-Games, the proofs begin with the simpler ones andprogress upwards in difficulty.
Have Fun!
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September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
Trig — Level Three
Angles in a Triangle
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
26 Trig or Treat
Special Trig Identities for AllThree Angles in a Triangle
A + B + C = 90° + +2 2 2A B C
= sin (90° −sin ( ))2C
2A + B
= cos ( )2A + B
= cos (90° −cos ( ))2C
2A + B
= sin2
A + B
= 180°
C = 180° − (A + B )
sin C = sin (180° − (A + B ))
= sin (A + B )
cos C = cos (180° − (A + B ))
= −cos (A + B )
sin (A + B + C ) = sin 180° = 0
cos (A + B + C ) = cos 180° = −1
A
B C
See Graphs on p. 42
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
Angles in a Triangle 27
Trig Identities Involving All Three Angles in a Triangle
Some special trig identities apply only when all three angles in a triangleare involved. For such identities, the additional constraint of
(A+B+C) = 180◦
is a critical one.For such identities the relationship between the three angles is always
necessary for simplification, and sometimes result in beautiful identities asseen in some of the examples in Level-Three-Games.
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
28 Trig or Treat
The Sum and Difference of Sine Functions
sin(A+B) = sin Acos B+ cos Asin B
sin(A−B) = sin Acos B− cos Asin B
Adding the two equations:
sin(A+B)+ sin(A−B) = 2 sin Acos B
sin S+ sin T = 2 sin
(
S+T2
)
cos
(
S−T2
)
Let (A+B) = S
and (A−B) = T
∴ A =S+T
2
and B =S−T
2
Subtracting the second equation from the first:
sin(A+B)− sin(A−B) = 2 cos A sin B
sin S− sin T = 2cos
(
S+T2
)
sin
(
S−T2
)
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
Angles in a Triangle 29
The Sum and Difference of Sine Functions
Often in Math, the addition or subtraction of similar equations, or the re-arrangement of sums and differences can lead to new insights and newequations which may be of special value.
The equations in the earlier pages are the sin and cos formulas of com-pound angles.
Here we are looking at the sum and difference of the trig functionsof such compound angles, and after further simplification, we derive newrelationships. Knowing these new relationships provide greater flexibilityand agility in the manipulation of the trig building blocks, and enables newequations or identities to be proved.
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
30 Trig or Treat
The Sum and Difference of Cosine Functions
cos(A+B) = cos A cos B− sin A sin B
cos(A−B) = cos A cos B+ sin A sin B
Adding the two equations:
cos(A+B)+ cos(A−B) = 2 cos A cos B
cos S+ cos T = 2 cos
(
S+T2
)
cos
(
S−T2
)
Let (A+B) = S
and (A−B) = T
∴ A =S+T
2
and B =S−T
2
Subtracting the second equation from the first:
cos(A+B)− cos(A−B) = −2 sin A sin B
cos S− cos T = −2 sin
(
S+T2
)
sin
(
S−T2
)
Note theminus signin front ofthe RHSterms
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
Angles in a Triangle 31
The Sum and Difference of Cosine Functions
Similar addition and subtraction of the cos functions for compound anglesgive similar equations for the sum and difference of cos functions.
While these functions were extremely useful before calculators andcomputers were available, they have fallen into disuse in modern timesexcept for purposes of “examinations”, to test the student’s versatility.
You are now ready to play Level-Three-Games — the “Not-So-Easy”Proofs.∗
Have Fun!
∗There are no simple sum and difference formulas for tan!
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
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September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
Practical Trig
Numerical Values of Special Angles
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
34 Trig or Treat
Numerical Values of Special Angles
11
22
30° 30°
60° 60°
3
3
3
3sin 30° =
2
1tan 30° =
1cos 30° =
2
31
sin 60° =2
tan 60° =cos 60° =2
45°
1
21
sin 45° = 2
1 cos 45° = 2
1 tan 45° = 11
90°
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
Numerical Values of Special Angles 35
Numerical Values of Special Angles
Five angles are of special interest in trigonometry — 0◦, 30◦, 45◦, 60◦, 90◦.Therefore it is important to know them and to remember them. The proofsare visual (resulting from the Pythagoras’ Theorem), and are easy to follow(see figures on the opposite page). Written in the form of square roots, theratios are easy to remember, beginning with sin 0◦ = 0, and cos 0◦ = 1.
sin 0◦ = 0 =
√
04
cos 0◦ = 1 =
√
44
tan 0◦ = 0
sin 30◦ =12
=
√
14
cos 30◦ =
√3
2=
√
34
tan 30◦ =1√
3
sin 45◦ =
√2
2=
√
24
cos 45◦ =
√2
2=
√
24
tan 45◦ = 1
sin 60◦ =
√3
2=
√
34
cos 60◦ =12
=
√
14
tan 45◦ =√
3
sin 90◦ = 1 =
√
44
cos 90◦ = 0 =
√
04
tan 90◦ (indeterminate)
Both sin 0◦ and tan 0◦ are zero because the length for “opposite” is zero.cos 0◦ = 1 because the “adjacent” and the hypotenuse are identical.
Similarly sin 90◦ = 1 because the “opposite” is coincident with the hy-potenuse; and cos 90◦ = 0 because the length of “adjacent” is zero.
It is important to stress than tan 90◦ does not have a value (indetermi-nate); in Math, we refer to it as “tending to infinity” and write it as “→ ∞”.
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
36 Trig or Treat
All Values of Sin (0◦ − 360◦)
Fourth QuadrantH
A4
O4
Second Quadrant H
A2 O2
Third Quadrant H
A3
O3
First Quadrant H
A1 O1
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
Numerical Values of Special Angles 37
Values of Obtuse and Negative Angles
Sine
One of the potential obstacles to having fun with Trig games is discoveringthat not all trig functions are positive. Trig functions of angles larger than90◦ (obtuse angles) can sometime have negative values.
One of the easiest ways to overcome these potential obstacles to havefun with Trig is to focus on the values of the sin function. Consider fourangles A1, A2, A3 and A4 in the four quadrants, respectively:
First Quadrant : sin A1 =O1
H∗=
+ve+ve
= +ve
Second Quadrant : sin A2 =O2
H=
+ve+ve
= +ve
Third Quadrant : sin A3 =O3
H=
−ve+ve
= −ve
Fourth Quadrant : sin A4 =O4
H=
−ve+ve
= −ve.
∗The hypotenuse always has a positive value.
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
38 Trig or Treat
Value of Sine in the Four Quadrants
1
090° 180° 270° 360°
−1
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
Numerical Values of Special Angles 39
Value of Sine in the Four Quadrants
The values of the sine of angles in the first and second quadrants (i.e.between 0◦ and 180◦) are always positive as seen in the graph on theopposite page, rising from 0 for sin 0◦ to a maximum value of 1 for sin 90◦.
Similarly the values of the sine of angles in the third and fourth quad-rants (i.e. between 180◦ and 360◦) are all negative, going to a minimum of−1 for sin 270◦, and returning to 0 for sine 360◦.
The brain remembers pictures better than equations or words; so com-mit the graph on the opposite page to memory; and this would prove to beextremely valuable in solving trig problems. This graph is well-known inMath as “the sinusoidal curve” or the “sine curve”, for short.
A quick sketch of the sinusoidal curve (done in 10 seconds) will providea good guide for ensuring that the correct values of sine in the differentquadrants are obtained.
It is also important to remember that the sine of a negative angle is thenegative of the sine of the angle.
sin(−A) = −sin A.
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
40 Trig or Treat
Value of Cosine in the Four Quadrants
1
090° 180° 270° 360°
−1
Value of Tangent in the Four Quadrants
0° 90° 180° 270° 360°
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
Numerical Values of Special Angles 41
Cosine and Tangent
By similar consideration, the values of the cos and tan of angles in thefour quadrants can be obtained. Again it is easier to remember the pictures(graphs on the opposite page).
Always remember that cos 0◦ = 1, so the graph for cos always beginsat 1, and go to −1 for cos 180◦. (The cos graph for 0◦ to 360◦ looks like ahole in the ground).
Again, remember:
cos(−A) = cos A.
The cos function has this unusual feature and hence cos functions needspecial attention (i.e. be extra careful with cos functions).
For tan, the value goes from tan 0◦ = 0 all the way to the indeterminatevalue for tan 90◦. Interestingly, the third quadrant is an exact replica of thefirst quadrant, and the fourth an exact replica of the second.
Again, remember:
tan(−A) = − tan A
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
42 Trig or Treat
(A) 90° 0°
0°
0°
cos
180° 270°
(180° + A)
(180° − A)
(180° − A)
(180° − A)
(180° + A)
(180° + A)
360°
90° 180° 270° 360°
90° 180° 270° 360°
(A)
(−A)
(−A)
(−A)
(A)
tan
sin
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3
Numerical Values of Special Angles 43
It is very difficult to remember all the equations for calculating the dif-ferent values of sin, cos and tan of angles in all four quadrants, as well asthe negative angles.
Fortunately, the three pictures (graphs on the opposite page) enable easyderivation of their values based on their relation to their primary values inthe first quadrant). Let us use an arbitrary angle (say 39◦) to see how wecan work out the correct values for angles in all the four quadrants.
sin(−39◦) = −sin 39◦ cos(−39◦) = cos 39◦
2nd Q sin 141◦ = sin(180◦−39◦) cos 141◦ = −cos(180◦−39◦)
= sin 39◦ = −cos 39◦
3rd Q sin(180◦ +39◦) = −sin 39◦ cos(180◦ +39◦) = −cos 39◦
4th Q sin(360◦−39◦) = −sin 39◦ cos(360◦−39◦) = cos 39◦
tan(−39◦) = − tan 39◦
2nd Q tan 141◦ = − tan(180◦−39◦)
= − tan 39◦
3rd Q tan(180◦ +39◦) = tan 39◦
4th Q tan(360◦−39◦) = − tan 39◦
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
Appendix
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
Concordance 45
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
46 Trig or Treat
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
The Concordanceof
Trigonometric Identities
The six-digit code for the Concordance is based on the number of trig functionson the LHS of the identity e.g. 123 000 means that the LHS has 1 sin, 2 cos, 3 tanand no cosec, sec and cot functions. On the rare occasion when you cannot findthe identity in the Concordance, use the functions on the RHS to determine thecode.
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
48 Trig or Treat
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
0 0 0 0 0 2 2 cot A cot 2A ≡ cot2 A−1 2 288
(1+ cot A)2 +(1− cot A)2≡
2
sin2 A1 233
cot4 A+ cot2 A ≡ cosec4 A− cosec2 A 1 188
cot A+1cot A−1
≡
1+ tan A1− tan A
1 110
1− cot2 A1+ cot2 A
≡ sin2 A− cos2 A 1 183
cot2 A−12 cot A
≡ cot 2A 2 315
0 0 0 0 0 4cot A cot B−1cot A+ cot B
≡ cot(A+B) 2 323
cot A cot B+1cot A− cot B
≡ cot(A−B) 2 274
0 0 0 0 1 0 2 sec2 A−1 ≡ 1+2tan2 A 1 201
1+2 sec2 A ≡ 2 tan2 A+3 1 128
0 0 0 0 1 1 cot A(sec2 A−1)≡ tan A 1 148
0 0 0 0 2 0 sec4−sec2 A ≡ tan4 A+ tan2 A 1 157
sec A1+ sec A
≡
1− cos A
sin2 A1 203
1+ sec A1− sec A
≡
cos A+1cos A−1
1 79
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
Concordance 49
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
0 0 0 0 2 0sec2 A
2− sec2 A≡ sec 2A 2 295
1+ sec Asec A
≡
sin2 A1− cos A
1 244
sec A+1sec A−1
≡ cot2A2
2 312
0 0 0 1 0 1 cosec2 A− cot2 A ≡ 1 1 145
cosec A− cot A ≡
sin A1+ cos A
1 219
cosec4 A− cot4 A ≡ cosec2 A+ cot2 A 1 103
cosec 2A− cot 2A ≡ tan A 2 299
cosec A−1cot A
≡
cot Acosec A+1
1 225
0 0 0 1 1 0 cosec A sec A ≡ 2cosec2A 2 270
cosec A sec A ≡ tan A+ cot A 1 135
cosec2 A+ sec2 A ≡ cosec2 A sec2 A 1 186
1sec2 A
+1
cosec2 A≡ 1 1 97
0 0 0 2 0 0cosec A−1cosec A+1
≡
1− sin A1+ sin A
1 109
cosec A1+ cosec A
≡
1− sin Acos2 A
1 215
cosec4A− cosec2A ≡ cot4 A+ cot2 A 1 158
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
50 Trig or Treat
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
0 0 0 2 0 2cosec A cosec Bcos A cot B−1
≡ sec(A+B) 2 273
0 0 0 2 2 0sec A
cosec2 A−
cosec Asec2 A
≡ (1+ cot A+ tan A)(sin A− cos A) 1 243
sec A− cosec Asec Acosec A
≡ sin A− cos A 1 113
0 0 1 0 0 0 tan2 A+1 ≡ sec2 A 1 146
tan
(
45◦ +A2
)
≡ tan A+ sec A 2 300
tan(
A−
π4
)
≡
tan A−1tan A+1
2 285
0 0 1 0 0 1 tan A+ cot A ≡ 2 cosec 2A 2 306
tan A+ cot A ≡ cosec A sec A 1 144
tan2 A+ cot2 A+2 ≡ cosec2 A sec2 A 1 101
tan A+ cot A ≡
2sin 2A
2 290
1+ cot A1+ tan A
≡ cot A 1 152
2cot A tan 2A
≡ 1− tan2 A 2 252
1tan A+ cot A
≡
sin Asec A
1 163
12(cot A− tan A) ≡ cot 2A 2 269
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
Concordance 51
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
0 0 1 0 0 11
tan A+ cot A≡ sin A cos A 1 166
0 0 1 0 0 2cot A(1+ tan2 A)
1+ cot2 A≡ tan A 1 180
0 0 1 0 1 0 sec 2A− tan 2A ≡ tan(45◦−A) 2 303
sec A− tan A ≡
cos A1+ sin A
1 218
sec4 A− tan4 A ≡
1+ sin2 Acos2 A
1 236
(sec A− tan A)2≡
1− sin A1+ sin A
1 230
tan Asec A−1
≡
sec A+1tan A
1 222
0 0 1 1 0 0 tan Acosec A ≡ sec A 1 104
tan2 A(cosec2 A−1)≡ 1 1 90
0 0 1 1 0 1 cosec A+ cot A+ tan A ≡
1+ cos Asin A cos A
1 131
cosec Acot A+ tan A
≡ cos A 1 159
0 0 1 1 1 0sec A+ cosec A
1+ tan A≡ cosec A 1 162
0 0 1 1 1 1 (tan A− cosec A)2− (cot A− sec A)2
≡ 2(cosec A− sec A) 2 338
0 0 2 0 0 0 tan A+ tan 2A ≡
sin A(4 cos2 A−1)
cos A cos 2A2 337
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
52 Trig or Treat
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
0 0 2 0 0 0 tan(45◦ +A) tan(45◦−A)
≡ cot(45◦ +A)cot(45◦−A) 2 326
(1+ tan A)2 +(1− tan A)2≡ 2 sec2 A 1 185
1− tan2 A1+ tan2 A
+1 ≡ 2 cos2 A 1 247
1− tan2 A1+ tan2 A
≡ cos 2A 2 317
2 tan A1+ tan2 A
≡ sin 2A 2 316
1+ tan A1− tan A
≡
cot A+1cot A−1
1 110
0 0 2 0 0 1tan A(1+ cot2 A)
1+ tan2 A≡ cot A 1 178
tan A(cot A+ tan A) ≡ sec2 A 1 89
0 0 2 0 0 2 (tan A+ cot A)2− (tan A− cot A)2
≡ 4 1 182
tan A+ tan Bcot A+ cot B
≡ tan A tan B 1 171
tan A− cot Atan A+ cot A
≡ sin2 A− cos2 A 1 189
cot A− tan Acot A+ tan A
≡ cos 2A 2 280
tan A− cot Atan A+ cot A
+1 ≡ 2 sin2 A 1 239
tan A− cot Atan A+ cot A
≡ 1−2 cos2 A 1 245
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
Concordance 53
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
0 0 2 0 0 2cot A
1− tan A+
tan A1− cot A
≡ 1+ tan A+ cot A 1 224
cot A1− tan A
+tan A
1− cot A≡ 1+ sec Acosec A 1 220
0 0 2 0 2 0 (sec A− tan A)(sec A+ tan A) ≡ 1 1 194
sec2 A− tan2 A+ tan Asec A
≡ sin A+ cos A 1 192
tan A+ sec A−1tan A− sec A+1
≡ tan A+ sec A 1 208
sec A sec B1+ tan A tan B
≡ sec(A−B) 2 272
0 0 2 1 2 0(sec A− tan A)2 +1
cosec A(sec A− tan A)≡ 2 tan A 1 249
0 0 3 0 0 0 tan A+ tan(A+120◦)+ tan(A+240◦)
≡ 3 tan 3A 3 391
tan A+ tan B+ tan C ≡ tan A tan B tan C(where A+B+C = 180◦) 3 356
3 tan A− tan3 A1−3 tan2 A
≡ tan 3A 2 325
tan Atan 2A− tan A
≡ cos 2A 2 302
0 0 4 0 0 01− tan A tan B1+ tan A tan B
≡
cos(A+B)
cos(A−B)2 277
tan A+ tan Btan A− tan B
≡
sin(A+B)
sin(A−B)2 278
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
54 Trig or Treat
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
0 0 4 0 0 01+ tan A1− tan A
+1− tan A1+ tan A
≡ 2 sec 2A 2 332
0 0 4 0 0 4 (tan A+ tan B)(1− cot A cot B)
+(cot A+ cot B)(1− tan A tan B) ≡ 0 1 114
0 1 0 0 0 0 2 cos2(45◦−A)≡ 1+ sin 2A 2 307
18(1− cos 4A) ≡ sin2 A cos2 A 2 314
1−2 cos2 A ≡ 2 sin2 A−1 1 149
21+ cos A
≡ sec2 A2
2 292
21− cos A
≡ cosec2 A2
2 291
0 1 0 0 0 21− cot2 A1+ cot2 A
+2 cos2 A ≡ 1 1 240
0 1 0 0 1 0 sec A− cos A ≡ sin A tan A 1 143
(1− cos A)(1+ sec A) ≡ sin A tan A 1 199
1+ sec A1+ cos A
≡ sec A 1 160
0 1 0 0 1 1 (cos A+ cot A)sec A ≡ 1+ cosec A 1 141
0 1 0 1 0 0 cos Acosec A ≡ cot A 1 83
cosec A1− cos A
≡
1+ cos A
sin3 A1 226
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
Concordance 55
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
0 1 0 1 0 1 cos2 A(cosec2 A− cot2 A) ≡ cos2 A 1 150
cot A cos Acosec2 A−1
≡ sin A 1 123
0 1 1 0 0 0 cos A tan A ≡ sin A 1 82
cos2 A(1+ tan2 A) ≡ 1 1 117
cos2 A tan A ≡ sin A cos A 1 86
(1+ cos A) tanA2≡ sin A 2 293
(cos2 A−1)(tan2 A+1)≡− tan2 A 1 122
1cos2 A(1+ tan2 A)
≡ 1 1 181
0 1 1 0 1 0sec A− cos A
tan A≡ sin A 1 207
0 1 1 0 1 1sec A+ tan Acot A+ cos A
≡ tan A sec A 1 210
0 1 1 1 0 0 cos A tan Acosec A ≡ 1 1 85
cosec Acot A+ tan A
≡ cos A 1 159
cos Acosec Atan A
≡ cot2 A 1 126
0 2 0 0 0 0 4 cos3 A−3 cos A ≡ cos 3A 2 340
cos 3A+ cos 2A ≡ 2 cos5A2
cosA2
3 358
8 cos4 A−4 cos 2A−3≡ cos 4A 2 345
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
56 Trig or Treat
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
0 2 0 0 0 0cos A+1cos A−1
≡
1+ sec A1− sec A
1 79
11− cos A
+1
1+ cos A≡ 2 cosec2A 1 176
1− cos A1+ cos A
≡ (cosec A− cot A)2 1 231
cos(A+B)cos(A−B)≡ cos2 A− sin2 B 2 308
cos 4A+4 cos 2A+3≡ 8 cos4 A 2 342
cos 2A− cos 10A
≡ tan 4A(sin 2A+ sin 10A) 3 383
0 2 0 0 0 1 cos2 A+ cot2 Acos2 A ≡ cot2 A 1 184
cos A+ cos A cot2 A ≡ cot A cosec A 1 147
0 2 0 0 1 0 cos A(sec A− cos A) ≡ sin2 A 1 121
1− sec2 A(1− cos A)(1+ cos A)
≡−sec2 A 1 179
0 2 0 0 2 0sec A− cos Asec A+ cos A
≡
sin2 A1+ cos2 A
1 190
0 2 0 2 0 0cos A
cosec A−1+
cos Acosec A+1
≡ 2 tan A 1 205
0 2 1 0 0 0 cos2 A+ tan2 A cos2 A ≡ 1 1 119
0 2 1 1 0 0 cosec A tanA2−
cos 2A1+ cos A
≡ 4 sin2 A2
2 328
0 3 0 0 0 0 32 cos6 A−48 cos4 A+18 cos2 A−1
≡ cos 6A 2 341
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
Concordance 57
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
0 3 0 0 0 0 cos 3A+2 cos 5A+ cos 7A
≡ 4 cos2 A cos 5A 3 385
cos(A−B)
cos A cos B≡ 1+ tan A tan B 2 255
1− cos 2A+ cos 4A− cos 6A
≡ 4 sin A cos 2A sin 3A 3 380
1+ cos 2A+ cos 4A+ cos 6A
≡ 4 cos A cos 2A cos 3A 3 386
cos(A+B)
cos A cos B≡ 1− tan A tan B 2 275
cos A+ cos 3A2 cos 2A
≡ cos A 3 368
cos A+ cos B+ cos C
≡ 4 sinA2
sinB2
sinC2
+1
(where A+B+C = 180◦) 3 357
0 4 0 0 0 0cos 4A− cos 8Acos 4A+ cos 8A
≡ tan 2A tan 6A 3 377
cos 2A− cos 4Acos 2A+ cos 4A
≡ tan 3A tan A 3 381
cos A+ cos Bcos A− cos B
≡−cot
(
A+B2
)
cot
(
A−B2
)
3 362
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
58 Trig or Treat
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
1 0 0 0 0 0 1−2 sin2 A ≡ 2cos2 A−1 1 155
sin(A+B+C)≡
sin A cos B cos C
+sin B cos C cos A
+sin C cos A cos B
−sin A sin B sin C
3 352
1 0 0 0 0 1 sin A cot A ≡ cos A 1 80
1 0 0 0 1 0 (1− sin2 A) sec2 A ≡ 1 1 88
sec A1− sin A
≡
1+ sin Acos3 A
1 242
sin A sec A ≡ tan A 1 81
1− sin Asec A
≡
cos3 A1+ sin A
1 237
1 0 0 1 0 0 cosec A− sin A ≡ cot A cos A 1 140
(1− sin A)(1+ cosec A) ≡ cos A cot A 1 200
1+ sin A1+ cosec A
≡ sin A 1 75
1 0 1 0 0 0tan Asin A
≡ sec A 1 105
sin A tanA2≡ 1− cos A 2 297
sin 2A tan A ≡ 1− cos 2A 2 298
tan A sin A ≡ sec A− cos A 1 197
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
Concordance 59
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
1 0 1 0 0 0 tan2 A− sin2 A ≡ tan2 A sin2 A 1 151
1 0 1 0 1 0 sec A− sin A tan A ≡ cos A 1 139
(1− sin A)(sec A+ tan A) ≡ cos A 1 196
sin A sec Atan A
≡ 1 1 91
1+ sec Atan A+ sin A
≡ cosec A 1 204
tan A sin Asec2 A−1
≡ cos A 1 127
1 1 0 0 0 0 3 sin2 A+4 cos2 A ≡ 3+ cos2 A 1 111
cos 2A1+ sin 2A
≡
cot A−1cot A+1
2 322
sin 2A1− cos 2A
≡ cot A 2 267
1−cos2 A
1+ sin A≡ sin A 1 193
cos4 A− sin4 A ≡ cos2 A− sin2 A 1 102
cos4 A− sin4 A ≡
1sec 2A
2 253
cos4 A− sin4 A ≡ cos 2A 2 260
sin A1+ cos A
≡ tanA2
2 262
sin 2A1+ cos 2A
≡ tan A 2 305
1− cos Asin A
≡ tanA2
2 263
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
60 Trig or Treat
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
1 1 0 0 0 0 cos2 A− sin2 A ≡ 2 cos2 A−1 1 118
1−sin2 A
1− cos A≡−cos A 1 174
1−sin2 A
1+ cos A≡ cos A 1 227
sin4 A+ cos4 A ≡
34
+14
cos 4A 2 344
(cos2 A−2)2−4 sin2 A ≡ cos4 A 1 153
sin(30◦+A)+ cos(60◦ +A)≡ cos A 2 265
cos2 A− sin2 A ≡ 1−2 sin2 A 1 87
cos2 A− sin2 A ≡ cos4 A− sin4 A 1 102
1−8 sin2 A cos2 A ≡ cos 4A 2 346
(
sinA2
+ cosA2
)2
≡ 1+ sin A 2 264
1−cos2 A
1+ sin A≡ sin A 1 168
1+cos2 A
sin A−1≡−sin A 1 232
cos2 A1− sin A
≡ 1+ sin A 1 125
1− cos Asin A
≡
sin A1+ cos A
1 165
1 1 0 0 0 1 sin A+ cos A cot A ≡ cosec A 1 138
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
Concordance 61
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
1 1 0 1 1 0 sin A cos Acosec A sec A ≡ 1 1 84
sin Acosec A
+cos Asec A
≡ 1 1 124
sec Acosec A
+sin Acos A
≡ 2 tan A 1 95
1 1 1 0 0 0 cos A+ sin A tan A ≡ sec A 1 137
tan A+cos A
1+ sin A≡ sec A 1 223
cos2 A− sin2 A1− tan2 A
≡ cos2 A 1 116
tan A+ cos Asin A
≡ sec A+ cot A 1 92
1 1 1 0 0 1 tan2 A cos2 A+ cot2 A sin2 A ≡ 1 1 112
tan A(sin A+ cot A cos A) ≡ sec A 1 198
sin2 A− tan Acos2 A− cot A
≡ tan2 A 1 241
1− sin2 A1− cos2 A
+ tan A cot A ≡ cosec2 A 1 130
cos A1− tan A
+sin A
1− cot A≡ sin A+ cos A 1 206
1 1 1 1 0 0 tan A cos A+ cosec A sin2 A ≡ 2 sin A 1 120
1 1 1 1 1 1 (cosecA− sin A)(sec A− cos A)
·(tan A+ cot A) ≡ 1 1 167
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
62 Trig or Treat
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
1 2 0 0 0 0sin3 A+ cos3 A
1−2 cos2 A≡
sec A− sin Atan A−1
1 191
2 sin(A−B)
cos(A+B)− cos(A−B)≡ cot A− cot B 2 321
(2 cos2 A−1)2
cos4 A− sin4 A≡ 1−2 sin2 A 1 246
sin A+ cos Acos A
≡ 1+ tan A 1 76
cos Acos A− sin A
≡
11− tan A
1 77
1−2 cos2 Asin A cos A
≡ tan A− cot A 1 175
cos(A+B)
sin A cos A≡ cot A− cot B 2 256
cos(A−B)
sin A cos B≡ cot A+ tan B 2 257
sin(A+B)
cos A cos B≡ tan A+ tan B 2 259
cos(A+B)
cos A sin B≡ cot B− tan A 2 271
5−10 cos2 Asin A− cos A
≡ 5(sin A+ cos A) 1 228
1 3 0 0 0 0cos A
1+ cos 2A+
sin A1− cos 2A
≡
sin A+ cos Asin 2A
2 331
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
Concordance 63
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
1 3 0 0 0 0sin A
(
4 cos2 A−1)
cos A cos 2A≡ tan A+ tan 2A 2 337
2 0 0 0 0 0 sin 5A− sin 3A ≡ 2 sin A cos 4A 3 359
3 sin A−4 sin3 A ≡ sin 3A 2 324
2 sin2 A6− sin2 A
7≡ cos2 A
7− cos
A3
2 304
sin2 A
1− sin2 A≡ tan2 A 1 100
1+ sin A1− sin A
≡ (sec A+ tan A)2 1 229
1+ sin A1− sin A
≡
cosec A+1cosec A−1
1 78
1− sin A1+ sin A
≡ (sec A− tan A)2 1 248
sin(A+B)sin(A−B)≡ cos2 B− cos2 A 2 311
1sin A+1
−
1sin A−1
≡ 2 sec2 A 1 177
2 sin 2A(1−2 sin2 A) ≡ sin 4A 2 339
sin(A+B)sin(A−B)≡ sin2 A− sin2 B 2 310
2 0 0 0 0 1 sin A+ sin Acot2 A ≡ cosec A 1 195
2 0 0 1 0 0 sin A(cosec A− sin A) ≡ cos2 A 1 134
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
64 Trig or Treat
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
2 0 1 0 0 0 tan 4A(sin 2A+ sin 10A)
≡ cos 2A− cos 10A 3 388
sin A+ tan Asin A
≡ 1+ sec A 1 94
sin A+ sin A tan2 A ≡ tan A sec A 1 154
2 1 0 0 0 0 (4 sin A cos A)(1−2 sin2 A) ≡ sin 4A 2 283
(
2 sin2 A−1)2
sin4 A− cos4 A≡ 1−2 cos2 A 1 136
sin2 A+ cos2 A
sin2 A≡ cosec2 A 1 99
sin Asin A− cos A
≡
11− cot A
1 93
1−2 sin2 Asin A cos A
≡ cot A− tan A 1 169
1+ sin A− sin2 Acos A
≡ cos A+ tan A 1 96
cos(A−B)
sin A sin B≡ cot A cot B+1 1 258
sin(A+B)
sin A cos B≡ cot A tan B+1 2 276
sin(A−B)
sin A cos B≡ 1− cot A tan B 2 254
2 1 0 0 1 0sec Asin A
−
sin Acos A
≡ cot A 1 213
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
Concordance 65
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
2 1 0 0 2 0 2 sec2 A−2 sec2 A sin2 A
−sin2 A− cos2 A ≡ 1 1 187
2 2 0 0 0 0 (cos A− sin A)2 +2 sin A cos A ≡ 1 1 133
(a sin A+b cos A)2 +(a cos A−b sin A)2
≡ a2 +b2 1 106
(2a sin A cos A)2 +a2(cos2 A− sin2 A)2≡ a2 2 261
sin 3Asin A
−
cos 3Acos A
≡ 2 2 287
1+ sin Acos A
+cos A
1+ sin A≡
2cos A
1 164
cos A
1− sin2 A− cos2 A+ sin A≡ cot A 1 129
sin A cos A
cos2 A− sin2 A≡
tan A1− tan2 A
1 98
sin(A+45◦)cos(A+45◦)
+cos(A+45◦)sin(A+45◦)
≡ 2 sec 2A 2 336
cos 2Asin A
+sin 2Acos A
≡ cosec A 2 320
sin2 2A+2 cos2A−1
sin2 2A+3 cos 2A−3≡
11− sec 2A
2 335
sin A1− cos A
+1− cos A
sin A≡ 2cosec A 1 221
cos A1+ sin A
+1+ sin A
cos A≡ 2 sec A 1 216
1− sin Acos A
+cos A
1− sin A≡ 2 sec A 1 214
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
66 Trig or Treat
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
2 2 0 0 0 01+ cos A
sin A+
sin A1+ cos A
≡ 2cosec A 1 212
sin A1+ cos A
+1+ cos A
sin A≡
2sin A
1 161
1− cos 2A+ sin Asin 2A+ cos A
≡ tan A 2 347
sin A+ sin 2A2+3 cos A+ cos 2A
≡ tanA2
2 327
sin A+ sin 2A1+ cos A+ cos 2A
≡ tan A 2 268
(sin A+ cos A)2 +(sin A− cos A)2≡ 2 1 156
sin(2A+B)+ sin Bcos(2A+B)+ cos B
≡ tan(A+B) 3 361
sin(A+B)− sin(A−B)
cos(A+B)+ cos(A−B)≡ tan B 3 370
sin 4A− sin 2Acos 4A+ cos 2A
≡ tan A 3 366
sin 2A+ cos 2A+1sin 2A+ cos 2A−1
≡
tan(45◦ +A)
tan A2 333
1+ sin 2A+ cos 2Asin A+ cos A
≡ 2 cos A 2 266
cos2 A
sin2 A+ cos2 A+ sin2 A ≡
1
sin2 A1 132
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
Concordance 67
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
2 2 0 0 0 01+ sin A+ cos A1+ sin A− cos A
≡
1+ cos Asin A
1 211
1+ cos A+ sin A1+ cos A− sin A
≡ sec A+ tan A 1 217
sin3 A+ cos3 Asin A+ cos A
≡ 1− sin A cos A 1 107
cos3 A− sin3 Acos A− sin A
≡
2+ sin 2A2
2 286
sin A− cos A+1sin A+ cos A−1
≡
sin A+1cos A
1 209
sin3 A+ cos3 Asin A+ cos A
≡ 1−12
sin 2A 2 294
sin 4A+ sin 2Acos 4A+ cos 2A
≡ tan 3A 3 373
cos A− cos 3Asin 3A− sin A
≡ tan 2A 3 374
cos A− cos 3Asin 3A+ sin A
≡ tan A 3 367
cos A− cos 5Asin A+ sin 5A
≡ tan 2A 3 371
sin 4A+ sin 8Acos 4A+ cos 8A
≡ tan 6A 3 375
sin 4A− sin 8Acos 4A− cos 8A
≡−cot 6A 3 376
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
68 Trig or Treat
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
2 2 0 0 0 0sin A+ sin Bcos A+ cos B
≡ tan
(
A+B2
)
3 363
sin A− sin Bcos A− cos B
≡−cot
(
A+B2
)
3 364
2 2 0 0 0 2sin A cos Bcos A sin B
(cot Acot B)+1 ≡
1
sin2 B1 234
3 0 0 0 0 0 sin 5A+2sin 3A+ sin A ≡ 4sin 3Acos2 A 3 379
sin 2A+ sin 4A− sin 6A
≡ 4 sin A sin 2Asin 3A 3 384
sin 2A+ sin 2B+ sin 2C
≡ 4 sin A sin Bsin C
(where A+B+C = 180◦) 3 354
sin A+ sin B+ sin C
≡ 4cosA2
cosB2
cosC2
(where A+B+C = 180◦) 3 350
sin A+ sin 3A2sin 2A
≡ cos A 3 365
sin A(sin 3A+ sin 5A)
≡ cos A(cos 3A− cos 5A) 3 389
sin A(sin A+ sin 3A)
≡ cos A(cos A− cos 3A) 3 390
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
Concordance 69
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
3 1 0 0 0 0cos A+ sin A− sin3 A
sin A
≡ cot A+ cos2 A 1 115
sin A− sin 3A
sin2 A− cos2 A≡ 2 sin A 3 369
3 1 0 2 0 1 cosec A(cosec A− sin A)
+
(
sin A− cos Asin A
)
+ cot A ≡ cosec2 A 1 235
3 2 0 0 0 0sin 3A cos A− sin A cos 3A
sin 2A≡ 1 2 284
3 3 0 0 0 0 (sin A+ cos B)2
+(cos B+ sin A)(cos B− sin A)
≡ 2cos B(sin A+ cos B) 1 108
(sin A− cos B)2
+(cos B+ sin A)(cos B− sin A)
≡−2cos B(sin A− cos B) 1 202
sin A+ cos Acos A
−
sin A− cos Asin A
≡ sec Acosec A 1 173
sin A+ sin 2A+ sin 3Acos A+ cos 2A+ cos 3A
≡ tan 2A 3 372
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix
70 Trig or Treat
sin
cos
tan
cose
cse
c
cot
Lev
el
Page
3 3 0 0 0 0sin A+ cos A
sin A−
cos A− sin Acos A
≡ cosec A sec A 1 170
3 6 0 0 0 0 sin A cos B cos C + sin B cos A cos C
+sin C cos A cos B
≡ sin A sin B sin C
(A+B+C = 180◦) 3 353
4 0 0 0 0 0sin A+ sin Bsin A− sin B
≡ tan
(
A+B2
)
cot
(
A−B2
)
3 360
sin 4A+ sin 8Asin 4A− sin 8A
≡−
tan 6Atan 2A
3 378
1+ sin A1− sin A
−
1− sin A1+ sin A
≡ 4 tan A sec A 1 172
4 0 2 0 0 0sin 2A+ sin 4Asin 2A− sin 4A
+tan 3Atan A
≡ 0 3 382
4 2 0 0 0 0sin 2A cos A−2cos 2A sin A
2 sin A− sin 2A≡ 2 cos2 A
22 330
4 4 0 0 0 0cos A− cos Bsin A+ sin B
+sin A− sin Bcos A+ cos B
≡ 0 2 281
cos A− sin Acos A+ sin A
+cos A+ sin Acos A− sin A
≡ 2 sec 2A 2 279
cos A+ sin Acos A− sin A
−
cos A− sin Acos A+ sin A
≡ 2 tan 2A 2 282
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
The Encyclopedia ofTrigonometric Games or
Trigonometric Identity Proofs(TIPs)
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
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September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-GamesEasy Proofs
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
74 Trig or Treat
An Example for Proving a Trig Identity
1+ sin A1+ cosec A
≡ sin A
Eyeballing and Mental Gymnastics
1. Start with the more complex side, normally the LHS; this is more amenable tosimplification.
2. Consider simplifying tan, cot, and the reciprocal functions to sin and cos.3. Consider the use of common denominators.4. Rearrange and simplify through cancellation of common terms, if available.
Let us explore this first game (proof) together. Eyeballing the identity,we see that the more complex side is indeed the LHS (occassionally theRHS is the more complex; then it may be preferable to begin with theRHS. On rare occassions both the LHS and the RHS are complex; thenone can explore simplifying both side to a common set of terms).
We note the cosec term in the denominator and remember that cosec =
1/sin.Generally speaking it is easier to work with sin and cos than with
their reciprocals as it makes rearrangement, simplification and cancellationeasier.
With a reciprocal term in the denominator, we expect to use commondenominators prior to rearrangement, simplification and cancellation.
If all goes well, and no careless mistakes are made, we should end upwith “sin A” which is the target objective on the RHS.
It is useful to begin a proof by writing down accurately the LHS. If amistake is made here, no amount of effort will give the required identity.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 75
1+ sin A1+ cosec A
≡ sin A
LHS =1+ sin A
1+ cosec A
=1+ sin A
(
1+1
sin A
)
= (1+ sin A)
1sin A+1
sin A
= (1+ sin A)
(
sin Asin A+1
)
= sin A
≡ RHS.
Use a bracket, ifit helps focus onthe key groups andminimises carelessmistakes.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
76 Trig or Treat
sin A+ cos Acos A
≡ 1+ tan A
Eyeballing and Mental Gymnastics∗
1. t = s/c2. rearrange and simplify.
LHS =sin A+ cos A
cos A
= tan A+1
≡ RHS.
∗Some teachers prefer that students always write out the trig functions to includethe angle i.e. sin A instead of sin. In the “Eyeballing and Mental Gymnastics”sections, and in the short explanatory notes, we will use the abbreviations: s = sin,c = cos, and t = tan. Such abbreviations reflect the mental process in action, andconveys a sense of speed with eyeballing and mental gymnastics taking place.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 77
cos Acos A− sin A
≡
11− tan A
Eyeballing and Mental Gymnastics
1. divide LHS by cos2. t = s/c
LHS =cos A
cos A− sin A
=1
1− tan A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
78 Trig or Treat
1+ sin A1− sin A
≡
cosec A+1cosec A−1
Eyeballing and Mental Gymnastics
1. cosec = 1/sin2. rearrange and simplify.
LHS =1+ sin A1− sin A
=cosec +1cosec−1
≡ RHS.
divide bothnumeratorand denominatorby sin
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 79
cos A+1cos A−1
≡
1+ sec A1− sec A
Eyeballing and Mental Gymnastics
1. sec = 1/cos2. rearrange and simplify.
LHS =cos A+1cos A−1
=1+ sec A1− sec A
≡ RHS.
divide bothnumeratorand denominatorby cos
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
80 Trig or Treat
sin A cot A ≡ cos A
Eyeballing and Mental Gymnastics
1. cot = c/s2. simplify.
LHS = sin A cot A
= sin A ·
cos Asin A
= cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 81
sin A sec A ≡ tan A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, t = s/c2. simplify.
LHS = sin A · sec A
= sin A ·
1cos A
= tan A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
82 Trig or Treat
cos A tan A ≡ sin A
Eyeballing and Mental Gymnastics
1. t = s/c2. simplify.
LHS = cos A tan A
= cos Asin Acos A
= sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 83
cos A cosec A ≡ cot A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, cot = c/s2. simplify.
LHS = cos A cosec A
= cos A ·
1sin A
= cot A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
84 Trig or Treat
sin A cos A cosec A sec A ≡ 1
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, sec = 1/cos2. simplify.
LHS = sin A cos A cosec A sec A
= sin A cos A ·
1sin A
·
1cos A
= 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 85
cos A tan A cosec A ≡ 1
Eyeballing and Mental Gymnastics
1. t = s/c, cosec = 1/sin2. simplify.
LHS = cos A tan A cosec A
= cos A ·
sin Acos A
·
1sin A
= 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
86 Trig or Treat
tan A cos2 A ≡ sin A cos A
Eyeballing and Mental Gymnastics
1. t = s/c2. simplify.
LHS = tan A cos2 A
=sin Acos A
· cos2 A
= sin A cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 87
cos2 A− sin2 A ≡ 1−2 sin2 A
Eyeballing and Mental Gymnastics
1. c2, s2 suggest s2 + c2 ≡ 1∗
2. simplify.
LHS = cos2 A− sin2 A
= (1− sin2 A)− sin2 A
= 1−2 sin2 A
≡ RHS.
∗We shall use the abbreviation: s2 + c2 ≡ 1 for the trig equivalent of Pythagoras’Theorem sin2 A+ cos2 A ≡ 1.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
88 Trig or Treat
(1− sin2 A)sec2 A ≡ 1
Eyeballing and Mental Gymnastics
1. s2 suggests s2 + c2 ≡ 12. sec = 1/cos3. simplify.
LHS = (1− sin2 A)sec2 A
= cos2 A ·
1cos2 A
= 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 89
tan A(cot A+ tan A) ≡ sec2 A
Eyeballing and Mental Gymnastics
1. t = s/c, cot = c/s2. sec2 suggests s2 + c2 = 1∗
3. simplify.
LHS = tan A(cot A+ tan A)
= 1+ tan2 A
= sec2 A
≡ RHS.
∗The identity s2 + c2 ≡ 1 should also trigger off possible reference to:
tan2 A+1 ≡ sec2 A (divide s2 + c2 ≡ 1 by c2)
and1+ cot2 A ≡ cosec2A (divide s2 + c2 ≡ 1 by s2).
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
90 Trig or Treat
tan2 A(cosec2 A−1) ≡ 1
Eyeballing and Mental Gymnastics
1. tan2, cosec2 suggest s2 + c2 ≡ 12. rearrange and simplify.
LHS = tan2 A(cosec2 A−1)
= tan2 A(cot2 A)
≡ 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 91
sin Asec Atan A
≡ 1
Eyeballing and Mental Gymnastics
1. sec = 1/cos, t = s/c2. simplify.
LHS =sin A sec A
tan A
= sin A ·
1cos A
·
cos Asin A
≡ 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
92 Trig or Treat
tan A+ cos Asin A
≡ sec A+ cot A
Eyeballing and Mental Gymnastics
1. t = s/c, sec = 1/cos, cot = c/s2. simplify.
LHS =tan A+ cos A
sin A
=sin Acos A
·
1sin A
+cos Asin A
= sec A+ cot A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 93
sin Asin A− cos A
≡
11− cot A
Eyeballing and Mental Gymnastics
1. cot = c/s2. rearrange and simplify.
LHS =sin A
sin A− cos A
=1
1−cos Asin A
=1
1− cot A
≡ RHS.
divide bothnumeratorand denominatorby sin A
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
94 Trig or Treat
sin A+ tan Asin A
≡ 1+ sec A
Eyeballing and Mental Gymnastics
1. t = s/c, sec = 1/cos2. simplify.
LHS =sin A+ tan A
sin A
=sin Asin A
+sin Acos A
·
1sin A
= 1+ sec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 95
sec Acosec A
+sin Acos A
≡ 2 tan A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, cosec = 1/sin, t = s/c2. rearrange and simplify.
LHS =sec A
cosec A+
sin Acos A
=1
cos A·
sin A1
+sin Acos A
= tan A+ tan A
= 2 tan A.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
96 Trig or Treat
1+ sin A− sin2 Acos A
≡ cos A+ tan A
Eyeballing and Mental Gymnastics
1. s2 suggests s2 + c2 ≡ 12. t = s/c3. rearrange and simplify.
LHS =1+ sin A− sin2 A
cos A
=cos2 A+ sin A
cos A
= cos A+ tan A
= RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 97
1sec2 A
+1
cosec2 A≡ 1
Eyeballing and Mental Gymnastics
1. sec = 1/cos, cosec = 1/sin2. sec2, cosec2 suggest s2 + c2 ≡ 13. simplify.
LHS =1
sec2 A+
1cosec2 A
= cos2 A+ sin2 A
= 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
98 Trig or Treat
sin A cos A
cos2 A− sin2 A≡
tan A1− tan2 A
Eyeballing and Mental Gymnastics
1. t = s/c2. divide both numerator and denominator by cos2 A to give cos2 A/cos2 A = 13. rearrange and simplify.
LHS =sin Acos A
cos2 A− sin2 A
=
sin Acos Acos2 A
cos2 A− sin2 Acos2 A
=tan A
1− tan2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 99
sin2 A+ cos2 A
sin2 A≡ cosec2 A
Eyeballing and Mental Gymnastics
1. s2, c2 suggest s2 + c2 ≡ 12. cosec = 1/sin3. simplify.
LHS =sin2 A+ cos2 A
sin2 A
=1
sin2 A
= cosec2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
100 Trig or Treat
sin2 A
1− sin2 A≡ tan2 A
Eyeballing and Mental Gymnastics
1. s2, t2 suggest s2 + c2 ≡ 12. t = s/c.
LHS =sin2 A
1− sin2 A
=sin2 Acos2 A
= tan2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 101
tan2 A+ cot2 A+2 ≡ cosec2 A+ sec2 A
Eyeballing and Mental Gymnastics
1. tan2, cot2, cosec2, sec2 suggest s2 + c2 ≡ 12. rearrange and simplify.
LHS = tan2 A+ cot2 A+2
= (tan2 A+1)+(cot2 A+1)
= sec2 A+ cosec2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
102 Trig or Treat
cos4 A− sin4 A ≡ cos2 A− sin2 A
Eyeballing and Mental Gymnastics
1. cos4−sin4 suggest a4 −b4 = (a2 −b2)(a2 +b2)2. c2, s2 suggest s2 + c2 ≡ 13. rearrange and simplify.
LHS = cos4 A− sin4 A
= (cos2 A− sin2 A)(cos2 A+ sin2 A)
= (cos2 A− sin2 A)(1)
= cos2 A− sin2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 103
cosec4 A− cot4 A ≡ cosec2 A+ cot2 A
Eyeballing and Mental Gymnastics
1. (cosec4 − cot4) suggest a4 −b4 = (a2 −b2)(a2 +b2)2. squares suggest s2 + c2 ≡ 13. rearrange and simplify.
LHS = cosec4 A− cot4 A
= (cosec2 A− cot2 A)(cosec2 A+ cot2 A)
= (1)(cosec2 A+ cot2 A)
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
104 Trig or Treat
tan A cosec A ≡ sec A
Eyeballing and Mental Gymnastics
1. t = s/c, cosec = 1/sin, sec = 1/cos2. simplify.
LHS = tan A cosec A
=sin Acos A
·
1sin A
=1
cos A
= sec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 105
tan Asin A
≡ sec A
Eyeballing and Mental Gymnastics
1. t = s/c2. simplify.
LHS =tan Asin A
=sin Acos A
·
1sin A
=1
cos A
= sec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
106 Trig or Treat
(a sin A+b cos A)2 +(a cos A−b sin A)2≡ a2 +b2
Eyeballing and Mental Gymnastics
1. ( )2 suggests expansion2. s2 + c2 ≡ 13. rearrange and simplify.
LHS = (a sin A+b cos A)2 +(a cos A−b sin A)2
= a2 sin2 A+2ab sin A cos A+b2 cos2 A
+a2 cos2 A−2ab sin Acos A+b2 sin2 A
= a2(sin2 A+ cos2 A)+b2(sin2 A+ cos2 A)
= a2 +b2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 107
sin3 A+ cos3 Asin A+ cos A
≡ 1− sin A cos A
Eyeballing and Mental Gymnastics
1. s3 + c3 suggests (s+ c)(s2 − cs+ c2)2. s2 + c2 ≡ 13. rearrange and simplify.
LHS =sin3 A+ cos3 Asin A+ cos A
=(sin A+ cos A)(sin2 A− sin A cos A+ cos2 A)
(sin A+ cos A)
= sin2 A− sin A cos A+ cos2 A
= 1− sin A cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
108 Trig or Treat
(sin A+cos B)2+(cos B+sin A)(cos B−sin A)≡2 cos B(sin A+cos B)
Eyeballing and Mental Gymnastics
1. Although the identity looks lengthy and complicated, closer inspection showsthat the term (sin A+ cos B) is common to both sides of the equation.
2. rearrange and simplify.
LHS = (sin A+ cos B)2 +(cos B+ sin A)(cos B− sin A)
= (sin A+ cos B)[(sin A+ cos B)+(cos B− sin A)]
= (sin A+ cos B)(2cos B)
= 2cos B(sin A+ cos B)
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 109
cosec A−1cosec A+1
≡
1− sin A1+ sin A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin2. rearrange and simplify.
LHS =cosec A−1cosec A+1
=
1sin A
−1
1sin A
+1
=1− sin A
sin A·
sin A1+ sin A
=1− sin A1+ sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
110 Trig or Treat
1+ tan A1− tan A
≡
cot A+1cot A−1
Eyeballing and Mental Gymnastics
1. t = s/c, cot = c/s2. rearrange and simplify.
LHS =1+ tan A1− tan A
=1+
sin Acos A
1−sin Acos A
=cos A+ sin A
cos A·
cos Acos A− sin A
=cos A+ sin Acos A− sin A
=cot A+1cot A−1
≡ RHS.
divide bothnumeratorand denominatorby sin
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 111
3 sin2 A+4 cos2 A ≡ 3+ cos2 A
Eyeballing and Mental Gymnastics
1. s2 + c2 ≡ 12. rearrange and simplify.
LHS = 3 sin2 A+4 cos2 A
= 3 sin2 A+3 cos2 A+ cos2 A
= 3(sin2 A+ cos2 A)+ cos2 A
= 3+ cos2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
112 Trig or Treat
tan2 A cos2 A+ cot2 A sin2 A ≡ 1
Eyeballing and Mental Gymnastics
1. t = s/c, cot = c/s2. s2 + c2 ≡ 13. rearrange and simplify.
LHS = tan2 A cos2 A+ cot2 A sin2 A
=sin2 Acos2 A
· cos2 A+cos2 A
sin2 A· sin2 A
= sin2 A+ cos2 A
= 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 113
sec A− cosec Asec A cosec A
≡ sin A− cos A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, cosec = 1/sin2. rearrange and simplify.
LHS =sec A− cosec Asec A cosec A
=sec A
sec A cosec A−
cosec Asec A cosec A
=1
cosec A−
1sec A
= sin A− cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
114 Trig or Treat
(tan A+ tan B)(1− cot A cot B)+(cot A+ cot B)(1− tan A tan B) ≡ 0
Eyeballing and Mental Gymnastics
1. cot = 1/ tan∗
2. expansion of factors3. rearrange and simplify.
LHS = (tan A+ tan B)(1− cot A cot B)+(cot A+ cot B)(1− tan A tan B)
= tan A+ tan B− tan A cot A cot B− tan B cot A cot B
+ cot A+ cot B− cot A tan A tan B− cot B tan A tan B
= tan A+ tan B− cot B− cot A
+ cot A+ cot B− tan B− tan A
= 0
≡ RHS.
∗Since the terms in the identity to be proved are all tan and cot, it is easier to thinkof cot as 1/tan rather than to convert tan and cot into sin and cos.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 115
cos A+ sin A− sin3 Asin A
≡ cot A+ cos2 A
Eyeballing and Mental Gymnastics
1. sin A− sin3 A suggests sin A(1− sin2 A)2. s2 + c2 ≡ 13. cot = c/s4. rearrange and simplify.
LHS =cos A+ sin A− sin3 A
sin A
=cos A+ sin A(1− sin2 A)
sin A
=cos A+ sin A(cos2 A)
sin A
= cot A+ cos2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
116 Trig or Treat
cos2 A− sin2 A1− tan2 A
≡ cos2 A
Eyeballing and Mental Gymnastics
1. t = s/c2. s2 + c2 ≡ 13. rearrange and simplify.
LHS =cos2 A− sin2 A
1− tan2 A
=(cos2 A− sin2 A)
1−sin2 Acos2 A
= (cos2 A− sin2 A)
(
cos2 A
cos2 A− sin2 A
)
= cos2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 117
cos2 A(1+ tan2 A) ≡ 1
Eyeballing and Mental Gymnastics
1. c2, t2 suggest s2 + c2 = 12. t = s/c3. rearrange and simplify.
LHS = cos2 A(1+ tan2 A)
= cos2 A+ cos2 A ·
sin2 Acos2 A
= cos2 A+ sin2 A
= 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
118 Trig or Treat
cos2 A− sin2 A ≡ 2cos2 A−1
Eyeballing and Mental Gymnastics
1. c2, s2 suggest s2 + c2 ≡ 12. simplify.
LHS = cos2 A− sin2 A
= cos2 A− (1− cos2 A)
= cos2 A−1+ cos2 A
= 2cos2 A−1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 119
cos2 A+ tan2 Acos2 A ≡ 1
Eyeballing and Mental Gymnastics
1. c2, t2 suggest s2 + c2 ≡ 12. t = s/c3. rearrange and simplify.
LHS = cos2 A+ tan2 Acos2 A
= cos2 A+sin2 Acos2 A
· cos2 A
= cos2 A+ sin2 A
= 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
120 Trig or Treat
tan A cos A+ cosec A sin2 A ≡ 2 sin A
Eyeballing and Mental Gymnastics
1. t = s/c, cosec = 1/sin2. rearrange and simplify.
LHS = tan A cos A+ cosec A sin2 A
=sin Acos A
· cos A+1
sin A· sin2 A
= sin A+ sin A
= 2 sin A
= RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 121
cos A(sec A− cos A) ≡ sin2 A
Eyeballing and Mental Gymnastics
1. sec = 1/cos2. rearrange and simplify.
LHS = cos A(sec A− cos A)
= cos A
(
1cos A
− cos A
)
= 1− cos2 A
= sin2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
122 Trig or Treat
(cos2 A−1)(tan2 A+1) ≡− tan2 A
Eyeballing and Mental Gymnastics
1. c2, t2 suggest s2 + c2 ≡ 12. t = s/c3. rearrange and simplify.
LHS = (cos2 A−1)(tan2 A+1)
= −sin2 A · sec2A
= −
sin2 Acos2 A
= − tan2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 123
cot A cos Acosec2
−1≡ sin A
Eyeballing and Mental Gymnastics
1. cosec2 suggests s2 + c2 ≡ 12. cot = c/s3. rearrange and simplify.
LHS =cot A cos A(cosec2
−1)
=cos Asin A
· cos A
(
1cot2 A
)
=cos Asin A
· cos A ·
sin2 Acos2 A
= sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
124 Trig or Treat
sin Acosec A
+cos Asec A
≡ 1
Eyeballing and Mental Gymnastics
1. cosec = 1/sin; sec = 1/cos2. rearrange and simplify.
LHS =sin A
cosec A+
cos AsecA
= sin A · sin A+ cos A · cos A
= sin2 A+ cos2 A
= 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 125
cos2 A1− sin A
≡ 1+ sin A
Eyeballing and Mental Gymnastics
1. c2 suggests s2 + c2 ≡ 12. rearrange and simplify.
LHS =cos2 A
1− sin A
=1− sin2 A1− sin A
=(1+ sin A)(1− sin A)
(1− sin A)
= 1+ sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
126 Trig or Treat
cos A cosec Atan A
≡ cot2 A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, t = s/c, cot = c/s2. simplify.
LHS =cos A cosec A
tan A
= cos A ·
1sin A
·
cos Asin A
=cos2 A
sin2 A
= cot2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 127
tan A sin Asec2 A−1
≡ cos A
Eyeballing and Mental Gymnastics
1. sec2 suggest s2 + c2 ≡ 12. simplify.
LHS =tan Asin Asec2 A−1
=sin Acos A
· sin A ·
1tan2 A
=sin2 Acos A
·
cos2 A
sin2 A
= cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
128 Trig or Treat
1+2 sec2 A ≡ 2 tan2 A+3
Eyeballing and Mental Gymnastics
1. sec2, tan2 suggests s2 + c2 ≡ 12. rearrange and simplify.
LHS = 1+2 sec2 A
= 1+2(1+ tan2 A)
= 1+2+2 tan2 A
= 2 tan2 A+3
= RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 129
cos A
1− sin2 A− cos2 A+ sin A≡ cot A
Eyeballing and Mental Gymnastics
1. s2, c2 suggest s2 + c2 ≡ 12. cot = c/s3. rearrange and simplify.
LHS =cos A
1− sin2 A− cos2 A+ sin A
=cos A
1− (sin2 A+ cos2 A)+ sin A
=cos Asin A
= cotA
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
130 Trig or Treat
1− sin2 A1− cos2 A
+ tan A cot A ≡ cosec2 A
Eyeballing and Mental Gymnastics
1. s2, c2 suggest s2 + c2 ≡ 12. tan ·cot = 13. rearrange and simplify.
LHS =1− sin2 A1− cos2 A
+ tan A cot A
=cos2 A
sin2 A+1
= cot2 A+1
= cosec2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 131
cosec A+ cot A+ tan A ≡
1+ cos Asin Acos A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, cot = c/s, t = s/c2. rearrange and simplify.
LHS = cosec A+ cot A+ tan A
=1
sin A+
cos Asin A
+sin Acos A
=cos A+ cos2 A+ sin2 A
sin A cos A
=1+ cos A
sin Acos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
132 Trig or Treat
cos2 A
sin2 A+ cos2 A+ sin2 A ≡
1
sin2 A
Eyeballing and Mental Gymnastics
1. s2, c2 suggests s2 + c2 ≡ 12. rearrange and simplify.
LHS =cos2 A
sin2 A+ cos2 A+ sin2 A
=cos2 A
sin2 A+1
=cos2 A+ sin2 A
sin2 A
=1
sin2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 133
(cos A− sin A)2 +2 sin A cos A ≡ 1
Eyeballing and Mental Gymnastics
1. ( )2 suggests s2 + c2 ≡ 12. rearrange and simplify.
LHS = (cos A− sin A)2 +2 sin A cos A
= (cos2 A−2 sin A cos A+ sin2 A)+2 sin A cos A
= cos2 A+ sin2 A
= 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
134 Trig or Treat
sin A(cosec A− sin A) ≡ cos2 A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin2. c2 suggests s2 + c2 = 13. simplify.
LHS = sin A(cosec A− sin A)
= sin A
(
1sin A
− sin A
)
= 1− sin2 A
= cos2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 135
cosec A sec A ≡ tan A+ cot A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, sec = 1/cos, t = s/c, cot = c/s2. rearrange and simplify.
LHS = cosec Asec A
=1
sin A·
1cos A
=(sin2 A+ cos2 A)
sin Acos A
= tan A+ cot A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
136 Trig or Treat
(2 sin2 A−1)2
sin4 A− cos4 A≡ 1−2cos2 A
Eyeballing and Mental Gymnastics
1. sin4−cos4 suggests (a2 −b2)(a2 +b2)2. 2 sin2 A−1 equals 1−2 cos2 A3. rearrange and simplify.
LHS =(2 sin2 A−1)2
sin4 A− cos4 A
=(1−2 cos2 A)2
(sin2 A− cos2 A)(sin2 A+ cos2 A)
=(1−2 cos2 A)2
(1−2 cos2 A)(1)
= 1−2 cos2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 137
cos A+ tan A sin A ≡ sec A
Eyeballing and Mental Gymnastics
1. t = s/c, sec = 1/cos2. simplify.
LHS = cos A+ tan A sin A
= cos A+sin Acos A
· sin A
=cos2 A+ sin2 A
cos A
=1
cos A
= sec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
138 Trig or Treat
sin A+ cos A cot A ≡ cosec A
Eyeballing and Mental Gymnastics
1. cot = c/s, cosec = 1/sin2. rearrange and simplify.
LHS = sin A+ cos A cot A
= sin A+ cos A ·
cos Asin A
=sin2 A+ cos2 A
sin A
=1
sin A
= cosec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 139
sec A− sin A tan A ≡ cos A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, t = s/c2. rearrange and simplify.
LHS = sec A− sin A tan A
=1
cos A−
sin A · sin Acos A
=1− sin2 A
cos A
=cos2 Acos A
= cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
140 Trig or Treat
cosec A− sin A ≡ cot A cos A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, cot = c/s2. rearrange and simplify.
LHS = cosec A− sin A
=1
sin A− sin A
=1− sin2 A
sin A
=cos2 Asin A
= cot A · cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 141
(cos A+ cot A)sec A ≡ 1+ cosec A
Eyeballing and Mental Gymnastics
1. cot = c/s, sec = 1/cos, cosec = 1/sin2. rearrange and simplify.
LHS = (cos A+ cot A)sec A
=
(
cos A+cos Asin A
)
·
1cos A
= cos A
(
1+1
sin A
)
1cos A
= 1+1
sin A
= 1+ cosec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
142 Trig or Treat
cosec A− sin A ≡ cos A cot A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, cot = c/s2. rearrange and simplify.
LHS = cosec A− sin A
=1
sin A− sin A
=1− sin2 A
sin A
=cos2 Asin A
= cos A cot A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 143
sec A− cos A ≡ sin A tan A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, t = s/c2. rearrange and simplify.
LHS = sec A− cos A
=1
cos A− cos A
=1− cos2 A
cos A
=sin2 Acos A
= sin A tan A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
144 Trig or Treat
tan A+ cot A ≡ sec A cosec A
Eyeballing and Mental Gymnastics
1. t = s/c, cot = c/s, sec = 1/cos, cosec = 1/sin2. s2 + c2 ≡ 13. rearrange and simplify.
LHS = tan A+ cot A
=sin Acos A
+cos Asin A
=sin2 A+ cos2 A
sin A cos A
=1
sin A cos A
= sec A cosec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 145
cosec2 A− cot2 A ≡ 1
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, cot = c/s2. cosec2, cot2 suggest s2 + c2 ≡ 13. simplify.
LHS = cosec2 A− cot2 A
=1
sin2 A−
cos2 A
sin2 A
=1− cos2 A
sin2 A
=sin2 A
sin2 A
= 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
146 Trig or Treat
tan2 A+1 ≡ sec2 A
Eyeballing and Mental Gymnastics
1. tan2, sec2 suggest s2 + c2 ≡ 12. simplify.
LHS = tan2 A+1
=sin2 Acos2 A
+1
=sin2 A+ cos2 A
cos2 A
=1
cos2 A
= sec2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 147
cos A+ cos A cot2 A = cot A cosec A
Eyeballing and Mental Gymnastics
1. c2 suggests s2 + c2 ≡ 12. cosec = 1/sin3. rearrange and simplify.
LHS = cos A+ cos A cot2 A
= cos A(1+ cot2 A)
= cos A · cosec2A
= cos A ·
1
sin2 A
= cot A cosec A
= RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
148 Trig or Treat
cot A(sec2 A−1) ≡ tan A
Eyeballing and Mental Gymnastics
1. cot = c/s, sec = 1/cos, t = s/c2. sec2 suggests s2 + c2 ≡ 13. rearrange and simplify.
LHS = cot A(sec2 A−1)
=1
tan A(tan2 A)
= tan A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 149
1−2 cos2 A ≡ 2 sin2 A−1
Eyeballing and Mental Gymnastics
1. c2, s2, 1 suggest s2 + c2 ≡ 12. rearrange and simplify.
LHS = 1−2 cos2 A
= 1−2(1− sin2 A)
= 1−2+2 sin2 A
= 2 sin2 A−1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
150 Trig or Treat
cos2 A(cosec2 A− cot2 A) ≡ cos2 A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, cot = c/s2. cos2, cosec2, cot2 suggest s2 + c2 ≡ 13. rearrange and simplify.
LHS = cos2 A(cosec2 A− cot2 A)
= cos2 A
(
1
sin2 A−
cos2 A
sin2 A
)
= cos2 A
(
1− cos2 A
sin2 A
)
= cos2 A
(
sin2 A
sin2 A
)
= cos2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 151
tan2 A− sin2 A ≡ tan2 A sin2 A
Eyeballing and Mental Gymnastics
1. t = s/c2. t2, s2 suggest s2 + c2 ≡ 13. rearrange and simplify.
LHS = tan2 A− sin2 A
=sin2 Acos2 A
− sin2 A
=sin2 A− sin2 A cos2 A
cos2 A
=sin2 A(1− cos2 A)
cos2 A
= tan2 A · sin2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
152 Trig or Treat
1+ cot A1+ tan A
≡ cot A
Eyeballing and Mental Gymnastics
1. cot = c/s, t = s/c2. rearrange and simplify.
LHS =1+ cot A1+ tan A
=
(
1+cos Asin A
)
1
1+sin Acos A
=
(
sin A+ cos Asin A
)
·
(
cos Acos A+ sin A
)
=cos Asin A
= cot A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 153
(cos2 A−2)2−4 sin2 A ≡ cos4 A
Eyeballing and Mental Gymnastics
1. c2, s2 suggest s2 + c2 ≡ 12. rearrange and simplify.
LHS = (cos2 A−2)2−4 sin2 A
= (cos4 A−4 cos2 A+4)−4 sin2 A
= cos4 A−4(cos2 A+ sin2 A)+4
= cos4 A−4(1)+4
= cos4 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
154 Trig or Treat
sin A+ sin A tan2 A ≡ tan A sec A
Eyeballing and Mental Gymnastics
1. t2 suggests s2 + c2 ≡ 12. sec = 1/cos3. rearrange and simplify.
LHS = sin A+ sin A tan2 A
= sin A(1+ tan2 A)
= sin A · sec2 A
= sin A ·
1cos2 A
= tan A sec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 155
1−2 sin2 A ≡ 2 cos2 A−1
Eyeballing and Mental Gymnastics
1. s2, c2 suggest s2 + c2 ≡ 12. rearrange and simplify.
LHS = 1−2 sin2 A
= 1− sin2 A− sin2 A
= cos2 A− sin2 A
= cos2 A+(cos2 A−1)
= 2 cos2 A−1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
156 Trig or Treat
(sin A+ cos A)2 +(sin A− cos A)2≡ 2
Eyeballing and Mental Gymnastics
1. ( )2 suggests expansion2. s2 + c2 ≡ 13. rearrange and simplify.
LHS = (sin A+ cos A)2 +(sin A− cos A)2
= (sin2 A+2 sin A cos A+ cos2 A)
+(sin2 A−2 sin A cos A+ cos2 A)
= 2(sin2 A+ cos2 A)
= 2(1)
= 2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 157
sec4 A− sec2 A ≡ tan4 A+ tan2 A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, t = s/c2. s2 + c2 ≡ 13. rearange and simplify.
LHS = sec4 A− sec2 A
= sec2 A(sec2 A−1)
= sec2 A tan2 A
= (tan2 A+1) tan2 A
= tan4 A+ tan2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
158 Trig or Treat
cosec4 A− cosec2 A ≡ cot4 A+ cot2 A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, cot = c/s2. s2 + c2 ≡ 13. rearrange and simplify.
LHS = cosec4 A− cosec2 A
= cosec2 A(cosec2 A−1)
= cosec2 A(cot2A)
= (1+ cot2 A)(cot2 A)
= cot2 A+ cot4 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 159
cosec Acot A+ tan A
≡ cos A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, cot = c/s, t = s/c2. rearrange and simplify.
LHS =cosec A
cot A+ tan A
=1
sin A
1cos Asin A
+sin Acos A
=1
sin A
(
sin Acos A
cos2 A+ sin2 A
)
=1
sin A
(
sin A cos A1
)
= cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
160 Trig or Treat
1+ sec A1+ cos A
≡ sec A
Eyeballing and Mental Gymnastics
1. sec = 1/cos2. rearrange and simplify.
LHS =1+ sec A1+ cos A
=
(
1+1
cos A
)(
11+ cos A
)
=
(
cos A+1cos A
)(
11+ cos A
)
=1
cos A
= sec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 161
sin A1+ cos A
+1+ cos A
sin A≡
2sin A
Eyeballing and Mental Gymnastics
1. common denominator2. s2 + c2 ≡ 13. rearrange and simplify.
LHS =sin A
1+ cos A+
1+ cos Asin A
=sin2 A+(1+ cos A)2
(1+ cos A)(sin A)
=sin2 A+(1+2 cos A+ cos2 A)
(1+ cos A)(sin A)
=2+2 cos A
(1+ cos A)(sin A)
=2
sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
162 Trig or Treat
sec A+ cosec A1+ tan A
≡ cosec A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, cosec = 1/sin, t = s/c2. rearrange and simplify.
LHS =sec A+ cosec A
1+ tan A
=
(
1cos A
+1
sin A
)
1
1+sin Acos A
=(sin A+ cos A)
cos A sin A
(
cos Acos A+ sin A
)
=1
sin A
= cosec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 163
1tan A+ cot A
≡
sin Asec A
Eyeballing and Mental Gymnastics
1. t = s/c, cot = c/s, sec = 1/cos2. common denominator3. rearrange and simplify.
LHS =1
tan A+ cot A
=1
sin Acos A
+cos Asin A
=cos A sin A
sin2 A+ cos2 A
= cos A sin A
=sin Asec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
164 Trig or Treat
1+ sin Acos A
+cos A
1+ sin A≡
2cos A
Eyeballing and Mental Gymnastics
1. common denominator2. s2 + c2 ≡ 13. rearrange and simplify.
LHS =1+ sin A
cos A+
cos A1+ sin A
=(1+ sin A)2 + cos2 A
cos A(1+ sin A)
=1+2 sin A+ sin2 A+ cos2 A
cos A(1+ sin A)
=2+2 sin A
cos A(1+ sin A)
=2
cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 165
1− cos Asin A
≡
sin A1+ cos A
Eyeballing and Mental Gymnastics
1. 1 suggests s2 + c2 ≡ 12. rearrange and simplify.
LHS =1− cos A
sin A
=1− cos A
sin A·
(
1+ cos A1+ cos A
)
=1− cos2 A
sin A(1+ cos A)
=sin2 A
sin A(1+ cos A)
=sin A
1+ cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
166 Trig or Treat
1tan A+ cot A
≡ sin A cos A
Eyeballing and Mental Gymnastics
1. t = s/c, cot = c/s2. rearrange and simplify.
LHS =1
tan A+ cot A
=1
sin Acos A
+cos Asin A
=cos A sin A
sin2 A+ cos2 A
=sin A cos A
1
= sin A cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 167
(cosec A− sin A)(sec A− cos A)(tan A+ cot A) ≡ 1
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, sec = 1/cos, t = s/c, cot = c/s2. common denominator3. rearrange and simplify.
LHS = (cosec A− sin A)(sec A− cos A)(tan A+ cot A)
=
(
1sin A
− sin A
)(
1cos A
− cos A
)(
sin Acos A
+cos Asin A
)
=
(
(1− sin2 A)
sin A
)(
1− cos2 Acos A
)(
sin2 A+ cos2 Acos A sin A
)
=
(
cos2 Asin A
)(
sin2 Acos A
)(
1cos A sin A
)
= 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
168 Trig or Treat
1−cos2 A
1+ sin A≡ sin A
Eyeballing and Mental Gymnastics
1. Common denominator2. c2 suggests s2 + c2 = 13. rearrange and simplify.
LHS = 1−cos2 A
1+ sin A
=1+ sin A− cos2 A
1+ sin A
=sin2 A+ sin A
1+ sin A
=sin A(1+ sin A)
(1+ sin A)
= sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 169
1−2 sin2 Asin A cos A
≡ cot A− tan A
Eyeballing and Mental Gymnastics
1. cot = c/s, t = s/c2. s2 suggests s2 + c2 ≡ 13. rearrange and simplify.
LHS =1−2 sin2 Asin A cos A
=1− sin2 A− sin2 A
sin A cos A
=cos2 A− sin2 A
sin A cos A
=cos Asin A
−
sin Acos A
= cot A− tan A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
170 Trig or Treat
sin A+ cos Asin A
−
cos A− sin Acos A
≡ cosec A sec A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, sec A = 1/cos A2. common denominator3. s2 + c2 ≡ 14. rearrange and simplify.
LHS =sin A+ cos A
sin A−
cos A− sin Acos A
=(sin A cos A+ cos2 A)− (sin A cos A− sin2 A)
sin A cos A
=cos2 A+ sin2 A
sin A cos A
=1
sin A cos A
= cosec A sec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 171
tan A+ tan Bcot A+ cot B
≡ tan A tan B
Eyeballing and Mental Gymnastics
1. cot = 1/ tan2. since both LHS and RHS are in terms of tan and cot, it may be easier to leave
the terms in tan rather than convert them to sin and cos.3. common denominator4. rearrange and simplify.
LHS =tan A+ tan Bcot A+ cot B
=tan A+ tan B
1tan A
+1
tan B
=(tan A+ tan B)
(
tan B+ tan Atan A tan B
)
= (tan A+ tan B)
(
tan A tan Btan A+ tan B
)
= tan A tan B
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
172 Trig or Treat
1+ sin A1− sin A
−
1− sin A1+ sin A
≡ 4 tan A sec A
Eyeballing and Mental Gymnastics
1. t = s/c, sec = 1/cos2. s2 + c2 ≡ 13. rearrange and simplify.
LHS =1+ sin A1− sin A
−
1− sin A1+ sin A
=(1+ sin A)2
− (1− sin A)2
(1− sin A)(1+ sin A)
=(1+2 sin A+ sin2 A)− (1−2 sin A+ sin2 A)
1− sin2 A
=4 sin Acos2 A
= 4 tan A sec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 173
sin A+ cos Acos A
−
sin A− cos Asin A
≡ sec A cosec A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, cosec = 1/sin2. common denominator3. s2 + c2 ≡ 14. rearrange and simplify.
LHS =sin A+ cos A
cos A−
sin A− cos Asin A
=(sin2 A+ sin A cos A)− (sin A cos A− cos2 A)
cos A sin A
=sin2 A+ cos2 A
cos A sin A
=1
cos A sin A
= sec A cosec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
174 Trig or Treat
1−sin2 A
1− cos A≡−cos A
Eyeballing and Mental Gymnastics
1. s2 + c2 ≡ 12. rearrange and simplify.
LHS = 1−sin2 A
1− cos A
= 1−(1− cos2 A)
(1− cos A)
= 1−(1− cos A)(1+ cos A)
(1− cos A)
= 1− (1+ cos A)
= −cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 175
1−2 cos2 Asin A cos A
≡ tan A− cot A
Eyeballing and Mental Gymnastics
1. c2 suggests s2 + c2 ≡ 12. t = s/c, cot = c/s3. rearrange and simplify.
LHS =1−2 cos2 Asin A cos A
=(sin2 A+ cos2 A)−2 cos2 A
sin A cos A
=sin2 A− cos2 A
sin A cos A
=sin Acos A
−
cos Asin A
= tan A− cot A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
176 Trig or Treat
11− cos A
+1
1+ cos A≡ 2 cosec2 A
Eyeballing and Mental Gymnastics
1. common denominator2. cosec = 1/sin3. cosec2 suggests s2 + c2 ≡ 14. rearrange and simplify.
LHS =1
1− cos A+
11+ cos A
=1+ cos A+1− cos A(1− cos A)(1+ cos A)
=2
1− cos2 A
=2
sin2 A
= 2 cosec2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 177
1sin A+1
−
1sin A−1
≡ 2 sec2 A
Eyeballing and Mental Gymnastics
1. sec = 1/cos2. sec2 suggests s2 + c2 ≡ 13. rearrange and simplify.
LHS =1
sin A+1−
1sin A−1
=(sin A−1)− (sin A+1)
(sin A+1)(sin A−1)
=−2
sin2 A−1
=−2
−cos2 A
= 2 sec2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
178 Trig or Treat
tan A(1+ cot2 A)
1+ tan2 A≡ cot A
Eyeballing and Mental Gymnastics
1. cot2, tan2 suggest s2 + c2 ≡ 12. cot = c/s, t = s/c3. rearrange and simplify.
LHS =tan A(1+ cot2 A)
1+ tan2 A
=sin Acos A
(cosec2 A)
(sec2 A)
=sin Acos A
(
1
sin2 A
)
·
(
cos2 A1
)
=cos Asin A
= cot A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 179
1− sec2 A(1− cos A)(1+ cos A)
≡−sec2 A
Eyeballing and Mental Gymnastics
1. sec2 suggests s2 + c2 ≡ 12. sec = 1/cos3. rearrange and simplify.
LHS =1− sec2 A
(1− cos A)(1+ cos A)
=− tan2 A
(1− cos2 A)
= −
sin2 Acos2 A
·
(
1
sin2 A
)
= −
1cos2 A
= −sec2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
180 Trig or Treat
cot A(1+ tan2 A)
1+ cot2 A≡ tan A
Eyeballing and Mental Gymnastics
1. tan2, cot2 suggest s2 + c2 ≡ 12. cot = c/s, t = s/c3. rearrange and simplify.
LHS =cot A(1+ tan2 A)
(1+ cot2 A)
=cos Asin A
(sec2 A)
(
1cosec2 A
)
=cos Asin A
·
(
1cos2 A
)
·
(
sin2 A1
)
=sin Acos A
= tan A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 181
1cos2 A(1+ tan2 A)
≡ 1
Eyeballing and Mental Gymnastics
1. c2, t2 suggest s2 + c2 = 12. t = s/c3. common denominator4. rearrange and simplify.
LHS =1
cos2 A(1+ tan2 A)
=1
cos2 A(sec2 A)
=1
cos2 A
(
1cos2 A
)
= 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
182 Trig or Treat
(tan A+ cot A)2− (tan A− cot A)2
≡ 4
Eyeballing and Mental Gymnastics
1. cot = 1/ tan2. expansion of squares3. ( )2 suggests s2 + c2 ≡ 14. rearrange and simplify.
LHS = (tan A+ cot A)2− (tan A− cot A)2
= (tan2 A+2 tan A cot A+ cot2 A)− (tan2 A−2 tan A cot A+ cot2 A)
= 2tan A cot A+2 tan A cot A
= 2+2
= 4
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 183
1− cot2 A1+ cot2 A
= sin2 A− cos2 A
Eyeballing and Mental Gymnastics
1. cot2, s2, c2 suggest s2 + c2 ≡ 12. cot = c/s3. rearrange and simplify.
LHS =1− cot2 A1+ cot2 A
=1− cot2 Acosec2 A
=
(
1−cos2 A
sin2 A
)(
sin2 A1
)
=(sin2 A− cos2 A)(sin2 A)
sin2 A
= sin2 A− cos2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
184 Trig or Treat
cos2 A+ cot2 A cos2 A ≡ cot2 A
Eyeballing and Mental Gymnastics
1. cot = c/s2. c2, cot2 suggest s2 + c2 ≡ 13. rearrange and simplify.
LHS = cos2 A+ cot2 A cos2 A
= cos2 A(1+ cot2 A)
= cos2 A(cosec2 A)
= cos2 A ·
1
sin2 A
= cot2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 185
(1+ tan A)2 +(1− tan A)2≡ 2sec2A
Eyeballing and Mental Gymnastics
1. t2, sec2 suggest s2 + c2 ≡ 12. rearrange and simplify.
LHS = (1+ tan A)2 +(1− tan A)2
= (1+2 tan A+ tan2 A)+(1−2 tan A+ tan2 A)
= 2+2 tan2 A
= 2(1+ tan2 A)
= 2 sec2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
186 Trig or Treat
cosec2 A+ sec2 A ≡ cosec2 A sec2 A
Eyeballing and Mental Gymnastics
1. cosec2, sec2 suggest s2 + c2 ≡ 12. cosec = 1/sin, sec = 1/cos3. rearrange and simplify.
LHS = cosec2 A+ sec2 A
=1
sin2 A+
1cos2 A
=cos2 A+ sin2 A
sin2 A cos2 A
=1
sin2 A cos2 A
= cosec2 A sec2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 187
2 sec2 A−2 sec2 A sin2 A− sin2 A− cos2 A ≡ 1
Eyeballing and Mental Gymnastics
1. sec2, s2, c2 suggest s2 + c2 ≡ 12. sec = 1/cos3. rearrange and simplify.
LHS = 2 sec2 A−2 sec2 A sin2 A− sin2 A− cos2 A
= 2 sec2 A(1− sin2 A)− (sin2 A+ cos2 A)
= 21
cos2 A· (cos2 A)− (1)
= 2−1
= 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
188 Trig or Treat
cot4 A+ cot2 A ≡ cosec4 A− cosec2 A
Eyeballing and Mental Gymnastics
1. cot = c/s, cosec = 1/sin2. cot2, cosec2 suggest s2 + c2 ≡ 13. rearrange and simplify.
LHS = cot4 A+ cot2 A
= cot2 A(cot2 A+1)
= cot2 A(cosec2 A)
= (cosec2 A−1)(cosec2 A)
= cosec4 A− cosec2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 189
tan A− cot Atan A+ cot A
≡ sin2 A− cos2 A
Eyeballing and Mental Gymnastics
1. t = s/c, cot = c/s2. common denominator3. s2 + c2 ≡ 14. rearrange and simplify.
LHS =tan A− cot Atan A+ cot A
=
sin Acos A
−
cos Asin A
sin Acos A
+cos Asin A
=
sin2 A− cos2 Acos A sin A
sin2 A+ cos2 Acos A sin A
=sin2 A− cos2 A
sin2 A+ cos2 A
= sin2 A− cos2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
190 Trig or Treat
sec A− cos Asec A+ cos A
≡
sin2 A1+ cos2 A
Eyeballing and Mental Gymnastics
1. sec = 1/cos2. common denominator3. s2 + c2 ≡ 14. rearrange and simplify.
LHS =sec A− cos Asec A+ cos A
=
1cos A
− cos A
1cos A
+ cos A
=
1− cos2 Acos A
1+ cos2 Acos A
=1− cos2 A1+ cos2 A
=sin2 A
1+ cos2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 191
sin3 A+ cos3 A1−2 cos2 A
≡
sec A− sin Atan A−1
Eyeballing and Mental Gymnastics
1. (s3 + c3) suggests (s+ c)(s2 − sc+ c2)2. (1−2 cos2 A) suggests (s2 − c2) and (s+ c)(s− c)3. sec = 1/cos, t = s/c4. rearrange and simplify.
LHS =sin3 A+ cos3 A
1−2 cos2 A
=(sin A+ cos A)(sin2 A− sin A cos A+ cos2 A)
sin2 A− cos2 A
=(sin A+ cos A)(1− sin A cos A)
(sin A+ cos A)(sin A− cos A)
=1− sin A cos Asin A− cos A
=sec A− sin A
tan A−1
≡ RHS.
divide bothnumeratorand denominatorby cos A
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
192 Trig or Treat
sec2 A− tan2 A+ tan Asec A
≡ sin A+ cos A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, t = s/c2. s2 + c2 ≡ 13. rearrange and simplify.
LHS =sec2
− tan2 A+ tan Asec A
=(1+ tan2 A)− tan2 A+ tan A
sec A
=1+ tan A
sec A
=1
sec A+
sin Acos A
·
cos A1
= cos A+ sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 193
1−cos2 A
1+ sin A≡ sin A
Eyeballing and Mental Gymnastics
1. s2 + c2 ≡ 12. rearrange and simplify.
LHS = 1−cos2 A
1+ sin A
= 1−(1− sin2 A)
1+ sin A
= 1−(1+ sin A)(1− sin A)
(1+ sin A)
= 1− (1− sin A)
= sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
194 Trig or Treat
(sec A− tan A)(sec A+ tan A) ≡ 1
Eyeballing and Mental Gymnastics
1. (sec− tan)(sec+ tan) suggests (a−b)(a+b) = a2 −b2
2. s2 + c2 ≡ 13. rearrange and simplify.
LHS = (sec A− tan A)(sec A+ tan A)
= sec2− tan2 A
= (1+ tan2 A)− tan2 A
= 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 195
sin A+ sin A cot2 A ≡ cosec A
Eyeballing and Mental Gymnastics
1. cot2 suggests s2 + c2 ≡ 12. cosec = 1/sin3. rearrange and simplify.
LHS = sin A+ sin A cot2 A
= sin A(1+ cot2 A)
= sin Acosec2 A
= sin A ·
1
sin2 A
=1
sin A
= cosec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
196 Trig or Treat
(1− sin A)(sec A+ tan A) ≡ cos A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, t = s/c2. common denominator3. rearrange and simplify.
LHS = (1− sin A)(sec A+ tan A)
= (1− sin A)
(
1cos A
+sin Acos A
)
= (1− sin A)
(
1+ sin Acos A
)
=1− sin2 A
cos A
=cos2 Acos A
= cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 197
tan A sin A ≡ sec A− cos A
Eyeballing and Mental Gymnastics
1. t = s/c, sec = 1/cos2. rearrange and simplify.
LHS = tan A sin A
=sin Acos A
· sin A
=sin2 Acos A
=(1− cos2 A)
cos A
=1
cos A− cos A
= sec A− cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
198 Trig or Treat
tan A(sin A+ cot A cos A) ≡ sec A
Eyeballing and Mental Gymnastics
1. t = s/c, cot = c/s, cosec = 1/sin2. simplify.
LHS = tan A(sin A+ cot A cos A)
=sin Acos A
(
sin A+cos Asin A
· cos A
)
=sin2 Acos A
+ cos A
=sin2 A+ cos2 A
cos A
=1
cos A
= sec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 199
(1− cos A)(1+ sec A) ≡ sin A tan A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, t = s/c2. rearrange and simplify.
LHS = (1− cos A)(1+ sec A)
= (1− cos A)
(
1+1
cos A
)
= (1− cos A)
(
1+ cos Acos A
)
=1− cos2 A
cos A
=sin2 Acos A
= sin A tan A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
200 Trig or Treat
(1− sin A)(1+ cosec A) ≡ cos A cot A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, cot = c/s2. rearrange and simplify.
LHS = (1− sin A)(1+ cosec A)
= 1− sin A+ cosec A− sin A · cosec A
= cosec A− sin A
=1
sin A− sin A
=1− sin2 A
sin A
=cos2 Asin A
= cos A cot A
≡ RHS.
sin A · cosec A= 1
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 201
2 sec2 A−1 ≡ 1+2 tan2 A
Eyeballing and Mental Gymnastics
1. sec2, t2 suggest s2 + c2 ≡ 12. rearrange and simplify.
LHS = 2 sec2 A−1
= 2(1+ tan2 A)−1
= 2+2 tan2 A−1
= 1+2 tan2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
202 Trig or Treat
(sinA−cosB)2+(cosB+sinA)(cosB−sinA)=−2cosB(sinA−cosB)
Eyeballing and Mental Gymnastics
1. Although the identity looks lengthy and complicated, closer inspection showsthat the term (sin A− cos B) is common on both sides of the equation.
2. Rearrange and simplify.
LHS = (sin A− cos B)2 +(cos B+ sin A)(cos B− sin A)
= (sin A− cos B)2− (sin A− cos B)(cos B+ sin A)
= (sin A− cos B)((sin A− cos B)− (cos B− sin A))
= (sin A− cos B)(−2 cos B)
= −2 cos B(sin A− cos B)
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 203
sec A1+ sec A
≡
1− cos A
sin2 A
Eyeballing and Mental Gymnastics
1. sec = 1/cos2. common denominator3. s2 + c2 ≡ 14. rearrange and simplify.
LHS =sec A
1+ sec A
=1
cos A·
1
1+1
cos A
=1
cos A·
cos A(cos A+1)
=1
cos A+1
=
(
1− cos A1− cos A
)
·
1(1+ cos A)
=1− cos A1− cos2 A
=1− cos A
sin2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
204 Trig or Treat
1+ sec Atan A+ sin A
≡ cosec A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, t = s/c, cosec = 1/sin2. rearrange and simplify.
LHS =1+ sec A
tan A+ sin A
=
(
1+1
cos A
)
1sin Acos A
+ sin A
=
(
cos A+1cos A
)(
cos Asin A+ sin A cos A
)
=cos A+1
sin A(1+ cos A)
=1
sin A
= cosec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 205
cos Acosec A−1
+cos A
cosec +1≡ 2 tan A
Eyeballing and Mental Gymnastics
1. common denominator2. t = s/c, cosec = 1/sin3. rearrange and simplify.
LHS =cos A
cosec A−1+
cos Acosec A+1
=cos A(cosec A+1)+ cos A(cosec A−1)
(cosec A−1)(cosec A+1)
=(cot A+ cos A)+(cot A− cos A)
cosec2 A−1
=2 cot Acot2 A
|cosec2−1 = cot2
=2
cot A
= 2 tan A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
206 Trig or Treat
cos A1− tan A
+sin A
1− cot A≡ sin A+ cos A
Eyeballing and Mental Gymnastics
1. t = s/c, cot = c/s2. common denominator3. rearrange and simplify.
LHS =cos A
1− tan A+
sin A1− cot A
=cos A
(
1−sin Acos A
) +sin A
(
1−cos Asin A
)
= cos A ·
cos A(cos A− sin A)
+ sin A ·
sin A(sin A− cos A)
=cos2 A− sin2 Acos A− sin A
=(cos A+ sin A)(cos A− sin A)
(cos A− sin A)
= cos A+ sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 207
sec A− cos Atan A
≡ sin A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, t = s/c2. simplify.
LHS =sec A− cos A
tan A
=
(
1cos A
− cos A
)(
cos Asin A
)
=
(
1− cos2 Acos A
)(
cos Asin A
)
=1− cos2 A
sin A
=sin2 Asin A
= sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
208 Trig or Treat
tan A+ sec A−1tan A− sec A+1
≡ tan A+ sec A
Eyeballing and Mental Gymnastics
1. s2 + c2 ≡ 1tan2 +1 ≡ sec2
1 ≡ sec2− tan2
2. rearrange and simplify.
LHS =tan A+ sec A−1tan A− sec A+1
=tan A+ sec A− (sec2 A− tan2 A)
tan A− sec A+1
=(tan A+ sec A)− (sec A+ tan A)(sec A− tan A)
tan A− sec A+1
=(sec A+ tan A)(1− sec A+ tan A)
tan A− sec A+1
=(sec A+ tan A)(tan A− sec A+1)
(tan A− sec A+1)
= tan A+ sec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 209
sin A− cos A+1sin A+ cos A−1
≡
sin A+1cos A
Eyeballing and Mental Gymnastics
1. s2 + c2 ≡ 12. multiple terms in denominator; may have to use s2 + c2 = 1 to arrive at com-
mon factor with numerator3. rearrange and simplify.
LHS =sin A− cos A+1sin A+ cos A−1
=sin A− cos A+1sin A+ cos A−1
·
(
sin A+1sin A+1
)
=(sin A− cos A+1)(sin A+1)
(sin2 A+ sin A cos A− sin A)+(sin A+ cos A−1)
=(sin A− cos A+1)(sin A+1)
−cos2 A+ sin A cos A+ cos A
=(sin A− cos A+1)(sin A+1)
cos A(−cos A+ sin A+1)
=sin A+1
cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
210 Trig or Treat
sec A+ tan Acot A+ cos A
≡ tan A sec A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, t = s/c, cot = c/s2. common denominator3. rearrange and simplify.
LHS =sec A+ tan Acot A+ cos A
=
1cos A
+sin Acos A
cos Asin A
+ cos A
=
1+ sin Acos A
cos A+ cos A sin Asin A
=(1+ sin A)
cos A·
sin Acos A(1+ sin A)
=sin Acos A
·
1cos A
= tan A sec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 211
1+ sin A+ cos A1+ sin A− cos A
≡
1+ cos Asin A
Eyeballing and Mental Gymnastics
1. multiple term in denominator; may have to find common factor forcancellation.
2. s2 + c2 ≡ 13. rearrange and simplify.
LHS =1+ sin A+ cos A1+ sin A− cos A
=1+ sin A+ cos A1+ sin A− cos A
·
(
1+ cos A1+ cos A
)
=(1+ sin A+ cos A)(1+ cos A)
(1+ sin A− cos A)+(cos A+ sin A cos A− cos2 A)
=(1+ sin A+ cos A)(1+ cos A)
sin2 A+ sin A+ sin A cos A
=(1+ sin A+ cos A)(1+ cos A)
sin A(sin A+1+ cos A)
=1+ cos A
sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
212 Trig or Treat
1+ cos Asin A
+sin A
1+ cos A≡ 2 cosec A
Eyeballing and Mental Gymnastics
1. common denominator2. cosec = 1/sin3. rearrange and simplify.
LHS =1+ cos A
sin A+
sin A1+ cos A
=(1+ cos A)2 + sin2 A
sin A(1+ cos A)
=1+2 cos A+ cos2 A+ sin2 A
sin A(1+ cos A)
=1+2 cos A+1sin A(1+ cos A)
=2(1+ cos A)
sin A(1+ cos A)
=2
sin A
= 2 cosec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 213
sec Asin A
−
sin Acos A
≡ cot A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, cot = c/s2. common denominator3. rearrange and simplify.
LHS =sec Asin A
−
sin Acos A
=1
cos A·
1sin A
−
sin Acos A
=1− sin2 A
cos A sin A
=cos2 A
cos A sin A
=cos Asin A
= cot A
= RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
214 Trig or Treat
1− sin Acos A
+cos A
1− sin A≡ 2 sec A
Eyeballing and Mental Gymnastics
1. sec = 1/cos2. common denominator3. s2 + c2 ≡ 14. rearrange and simplify.
LHS =1− sin A
cos A+
cos A1− sin A
=(1− sin A)2 + cos2 A
cos A(1− sin A)
=(1− sin A)2 +(1− sin2 A)
cos A(1− sin A)
=(1− sin A)2 +(1− sin A)(1+ sin A)
cos A(1− sin A)
=(1− sin A)(1− sin A+1+ sin A)
cos A(1− sin A)
=2
cos A
= 2 sec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 215
cosec A1+ cosec A
≡
1− sin Acos2 A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin2. common denominator3. s2 + c2 ≡ 14. rearrange and simplify.
LHS =cosec A
1+ cosec A
=1
sin A
1
1+1
sin A
=1
sin A
(
sin Asin A+1
)
=1
1+ sin A
=1
(1+ sin A)
(
1− sin A1− sin A
)
=1− sin A
1− sin2 A
=1− sin A
cos2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
216 Trig or Treat
cos A1+ sin A
+1+ sin A
cos A≡ 2 sec A
Eyeballing and Mental Gymnastics
1. sec = 1/cos2. common denominator3. s2 + c2 ≡ 14. rearrange and simplify.
LHS =cos A
1+ sin A+
1+ sin Acos A
=cos2 A+(1+ sin A)2
cos A(1+ sin A)
=(1− sin2 A)+(1+ sin A)2
cos A(1+ sin A)
=(1+ sin A)(1− sin A)+(1+ sin A)2
cos A(1+ sin A)
=(1+ sin A)(1− sin A+1+ sin A)
cos A(1+ sin A)
=2
cos A
= 2 sec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 217
1+ cos A+ sin A1+ cos A− sin A
≡ sec A+ tan A
Eyeballing and Mental Gymnastics
1. multiple terms in denominator suggest use of common factor for cancellation2. sec+ tan in RHS suggest 1/cos+ sin/cos which gives (1+ sin)/cos3. s2 + c2 ≡ 14. rearrange and simplify.
LHS =1+ cos A+ sin A1+ cos A− sin A
=1+ cos A+ sin A1+ cos A− sin A
·
(
1+ sin A1+ sin A
)
=(1+ cos A+ sin A)(1+ sin A)
1+ cos A− sin A+ sin A+ sin A cos A− sin2 A
=(1+ cos A+ sin A)(1+ sin A)
cos2 A+ cos A+ sin A cos A
=(1+ cos A+ sin A)(1+ sin A)
cos A(cos A+1+ sin A)
=1+ sin A
cos A
= sec A+ tan A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
218 Trig or Treat
sec A− tan A ≡
cos A1+ sin A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, t = s/c2. s2 + c2 ≡ 13. rearrange and simplify.
LHS = sec A− tan A
=1
cos A−
sin Acos A
=1− sin A
cos A
=1− sin A
cos A
(
1+ sin A1+ sin A
)
=1− sin2 A
cos A(1+ sin A)
=cos2 A
cos A(1+ sin A)
=cos A
1+ sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 219
cosec A− cot A ≡
sin A1+ cos A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, cot = c/s2. s2 + c2 ≡ 13. rearrange and simplify.
LHS = cosec A− cot A
=1
sin A−
cos Asin A
=1− cos A
sin A
=1− cos A
sin A
(
1+ cos A1+ cos A
)
=1− cos2 A
sin A(1+ cos A)
=sin2 A
sin A(1+ cos A)
=sin A
1+ cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
220 Trig or Treat
cot A1− tan A
+tan A
1− cot A≡ 1+ sec A cosec A
Eyeballing and Mental Gymnastics
1. cot = c/s, t = s/c, sec = 1/cos, cosec = 1/sin2. rearrange and simplify.
LHS =cot A
1− tan A+
tan A1− cot A
=cos Asin A
1(
1−sin Acos A
) +sin Acos A
·
1(
1−cos Asin A
)
=cos Asin A
·
cos Acos A− sin A
+sin Acos A
sin A(sin A− cos A)
=cos2 A
sin A(cos A− sin A)−
sin2 Acos A(cos A− sin A)
=cos3 A− sin3 A
(cos A− sin A)(sin A cos A)
=(cos A− sin A)(cos2 A+ cos A sin A+ sin2 A)
(cos A− sin A)(sin A cos A)
=(1+ cos A sin A)
sin A cos A
= cosec A sec A+1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 221
sin A1− cos A
+1− cos A
sin A≡ 2 cosec A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin2. common denominator3. rearrange and simplify.
LHS =sin A
1− cos A+
1− cos Asin A
=sin2 A+(1− cos A)2
sin A(1− cos A)
=sin2 A+(1−2 cos A+ cos2 A)
sin A(1− cos A)
=sin2 A+ cos2 A+1−2 cos A
sin A(1− cos A)
=2−2 cos A
sin A(1− cos A)
=2(1− cos A)
sin A(1− cos A)
=2
sin A
= 2 cosec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
222 Trig or Treat
tan Asec A−1
≡
sec A+1tan A
Eyeballing and Mental Gymnastics
1. t = s/c, sec = 1/cos c2. s2 + c2 ≡ 13. common denominator4. rearrange and simplify.
LHS =tan A
sec A−1
=sin Acos A
11
cos A−1
=sin Acos A
(
cos A1− cos A
)
=sin A
1− cos A
=sin A
(1− cos A)
(
1+ cos A1+ cos A
)
=sin A+ sin A cos A
1− cos2 A
=sin A+ sin A cos A
sin2 A
=1+ cos A
sin A
=sec A+1
tan A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 223
tan A+cos A
1+ sin A≡ sec A
Eyeballing and Mental Gymnastics
1. t = s/c, sec = 1/cos2. common denominator3. rearrange and simplify.
LHS = tan A+cos A
1+ sin A
=sin Acos A
+cos A
1+ sin A
=sin A(1+ sin A)+ cos2 A
cos A(1+ sin A)
=sin A+ sin2 A+ cos2 A
cos A(1+ sin A)
=(1+ sin A)
cos A(1+ sin A)
=1
cos A
= sec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
224 Trig or Treat
cot A1− tan A
+tan A
1− cot A≡ 1+ tan A+ cot A
Eyeballing and Mental Gymnastics
1. cot = c/s, t = s/c2. common denominator3. s2 + c2 ≡ 14. rearrange and simplify.
LHS =cot A
1− tan A+
tan A1− cot A
=
cos Asin A
1−sin Acos A
+
sin Acos A
1−cos Asin A
=cos Asin A
·
cos A(cos A− sin A)
+sin Acos A
·
sin A(sin A− cos A)
=cos3 A− sin3 A
sin Acos A(cos A− sin A)
=(cos A− sin A)(cos2 A+ sin Acos A+ sin2 A)
(cos A− sin A)(sin Acos A)
= cot A+1+ tan A
= 1+ tan A+ cot A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 225
cosec A−1cot A
≡
cot Acosec A+1
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, cot = c/s2. s2 + c2 ≡ 13. rearrange and simplify.
LHS =cosec A−1
cot A
=
1sin A
−1
cos Asin A
=1− sin A
sin A·
sin Acos A
=1− sin A
cos A
=1− sin A
cos A
(
1+ sin A1+ sin A
)
=1− sin2 A
cos A(1+ sin A)
=cos2 A
cos A(1+ sin A)
=cos A
1+ sin A
=cot A
cosec A+1
≡ RHS.
divide bothnumeratorand denominatorby sin
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
226 Trig or Treat
cosec A1− cos A
≡
1+ cos A
sin3 A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin2. sin3 suggests s · s2, s2 + c2 = 13. rearrange and simplify.
LHS =cosec A
1− cos A
=1
sin A·
1(1− cos A)
=1
sin A·
1(1− cos A)
·
(
1+ cos A1+ cos A
)
=1+ cos A
sin A(1− cos2 A)
=1+ cos A
sin A(sin2 A)
=1+ cos A
sin3 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 227
1−sin2 A
1+ cos A≡ cos A
Eyeballing and Mental Gymnastics
1. s2 suggests s2 + c2 ≡ 12. rearrange and simplify.
LHS = 1−sin2 A
1+ cos A
=(1+ cos A)− sin2 A
1+ cos A
=cos2 A+ cos A
1+ cos A
=cos A(cos A+1)
1+ cos A
= cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
228 Trig or Treat
5−10 cos2 Asin A− cos A
≡ 5(sin A+ cos A)
Eyeballing and Mental Gymnastics
1. c2 suggests s2 + c2 ≡ 12. rearrange and simplify.
LHS =5−10 cos2 Asin A− cos A
=5(1−2 cos2 A)
sin A− cos A
=5(cos2 A+ sin2 A−2 cos2 A)
sin A− cos A
=5(sin2 A− cos2 A)
sin A− cos A
=5(sin A+ cos A)(sin A− cos A)
(sin A− cos A)
= 5(sin A+ cos A)
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 229
1+ sin A1− sin A
≡ (sec A+ tan A)2
Eyeballing and Mental Gymnastics
1. (sec+ tan)2 suggests s2 + c2 ≡ 12. sec = 1/cos, t = s/c3. rearrange and simplify.
LHS =1+ sin A1− sin A
=1+ sin A1− sin A
(
1+ sin A1+ sin A
)
=(1+ sin A)2
1− sin2 A
=(1+ sin A)2
cos2 A
=
(
1+ sin Acos A
)2
= (sec A+ tan A)2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
230 Trig or Treat
(sec A− tan A)2≡
1− sin A1+ sin A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, t = s/c2. ( )2 suggests s2 + c2 ≡ 13. common denominator4. rearrange and simplify.
LHS = (sec A− tan A)2
=
(
1cos A
−
sin Acos A
)2
=(1− sin A)2
cos2 A
=(1− sin A)2
(1− sin2 A)
=(1− sin A)(1− sin A)
(1− sin A)(1+ sin A)
=1− sin A1+ sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 231
1− cos A1+ cos A
≡ (cosec A− cot A)2
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, cot = c/s2. ( )2 suggests s2 + c2 ≡ 13. rearrange and simplify.
LHS =1− cos A1+ cos A
=(1− cos A)
(1+ cos A)·
(
1− cos A1− cos A
)
=(1− cos A)2
(1− cos2 A)
=(1− cos A)2
sin2 A
=
(
1sin A
−
cos Asin A
)2
= (cosec A− cot A)2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
232 Trig or Treat
1+cos2 A
sin A−1≡−sin A
Eyeballing and Mental Gymnastics
1. c2 suggests s2 + c2 ≡ 12. rearrange and simplify.
LHS = 1+cos2 A
sin A−1
=(sin A−1)+ cos2 A
sin A−1
=(sin A− sin2 A− cos2 A)+ cos2 A
sin A−1
=sin A− sin2 A
sin A−1
=sin A(1− sin A)
sin A−1
= −sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 233
(1+ cot A)2 +(1− cot A)2≡
2
sin2 A
Eyeballing and Mental Gymnastics
1. ( )2, s2 suggest s2 + c2 ≡ 12. cot = c/s3. rearrange and simplify.
LHS = (1+ cot A)2 +(1− cot A)2
= (1+2 cot A+ cot2 A)+(1−2 cot A+ cot2 A)
= 2+2 cot2 A
= 2(1+ cot2 A)
= 2 cosec2 A
=2
sin2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
234 Trig or Treat
sin A cos Bcos A sin B
(cot A cot B)+1 ≡
1
sin2 B
Eyeballing and Mental Gymnastics
1. cot = c/s2. s2 suggests s2 + c2 ≡ 13. rearrange and simplify.
LHS =sin A cos Bcos A sin B
(cot A cot B)+1
=sin A cos Bcos A sin B
·
cos Asin A
·
cos Bsin B
+1
=cos2 B
sin2 B+1
=cos2 B+ sin2 B
sin2 B
=1
sin2 B
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 235
cosec A(cosec A− sin A)+sin A− cos A
sin A+ cot A ≡ cosec2 A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, cot = c/s2. rearrange and simplify.
LHS = cosec A(cosec A− sin A)+sin A− cos A
sin A+ cot A
=1
sin A
(
1sin A
− sin A
)
+sin A− cos A
sin A+
cos Asin A
=1− sin2 A
sin A · sin A+
sin A− cos Asin A
+cos Asin A
=1− sin2 A+ sin2 A− cos A sin A+ cos A sin A
sin2 A
=1
sin2 A
= cosec2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
236 Trig or Treat
sec4 A− tan4 A ≡
1+ sin2 Acos2 A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, t = s/c2. sec4− tan4 suggests a4 −b4 = (a2 −b2)(a2 +b2)3. s2, c2 suggest s2 + c2 ≡ 14. rearrange and simplify.
LHS = sec4 A− tan4 A
=1
cos4 A−
sin4 Acos4 A
=1− sin4 A
cos4 A
=(1− sin2 A)(1+ sin2 A)
cos4 A
=(cos2 A)(1+ sin2 A)
cos4 A
=1+ sin2 A
cos2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 237
1− sin Asec A
≡
cos3 A1+ sin A
Eyeballing and Mental Gymnastics
1. sec = 1/cos2. cos3 suggests c · c2, s2 + c2 ≡ 13. rearrange and simplify.
LHS =1− sin A
sec A
= (1− sin A)cos A
= (1− sin A)cos A
(
1+ sin A1+ sin A
)
=(1− sin2 A)cos A
(1+ sin A)
=(cos2 A)cos A
1+ sin A
=cos3 A
1+ sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
238 Trig or Treat
(1− cos A)
(1+ cos A)≡ (cosec A− cot A)2
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, cot = c/s2. square on RHS suggests s2 + c2 ≡ 13. rearrange and simplify.
LHS =1− cos A1+ cos A
=1− cos A1+ cos A
·
(
1− cos A1− cos A
)
=(1− cos A)2
1− cos2 A
=(1− cos A)2
sin2 A
=
(
1sin
−
cos Asin A
)2
= (cosec A− cot A)2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 239
tan A− cot Atan A+ cot A
+1 ≡ 2 sin2 A
Eyeballing and Mental Gymnastics
1. t = s/c, cot = c/s2. common denominators3. s2 + c2 ≡ 14. rearrange and simplify.
LHS =tan A− cot Atan A+ cot A
+1
=
sin Acos A
−
cos Asin A
sin Acos A
+cos Asin A
+1
=
sin2 A− cos2 Acos A sin A
sin2 A+ cos2 Acos A sin A
+1
= sin2 A− cos2 A+(sin2 A+ cos2 A)
= 2 sin2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
240 Trig or Treat
1− cot2 A1+ cot2 A
+2 cos2 A ≡ 1
Eyeballing and Mental Gymnastics
1. cot = c/s2. s2 + c2 ≡ 13. rearrange and simplify.
LHS =1− cot2 A1+ cot2 A
+2 cos2 A
=1−
cos2 A
sin2 A
1+cos2 A
sin2 A
+2 cos2 A
=
sin2 A− cos2 A
sin2 Asin2 A+ cos2 A
sin2 A
+2 cos2 A
=sin2 A− cos2 A
1+2 cos2 A
= sin2 A+ cos2 A
= 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 241
sin2 A− tan Acos2 A− cot A
≡ tan2 A
Eyeballing and Mental Gymnastics
1. t = s/c, cot = c/s2. s2 + c2 ≡ 13. rearrange and simplify.
LHS =sin2 A− tan Acos2 A− cot A
=sin2 A−
sin Acos A
cos2 A−
cos Asin A
=sin2 A cos A− sin A
cos A·
sin Acos2 A sin A− cos A
=sin A(sin A cos A−1)
cos A·
sin Acos A(sin A cos A−1)
=sin2 Acos2 A
= tan2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
242 Trig or Treat
sec A1− sin A
≡
1+ sin Acos3 A
Eyeballing and Mental Gymnastics
1. sec = 1/cos2. s2 + c2 ≡ 13. rearrange and simplify.
LHS =sec A
1− sin A
=1
cos A·
1(1− sin A)
=1
cos A(1− sin A)·
(
1+ sin A1+ sin A
)
=1+ sin A
cos A(1− sin2 A)
=1+ sin A
cos A(cos2 A)
=1+ sin A
cos3 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 243
sec Acosec2 A
−
cosec Asec2 A
≡ (1+ cot A+ tan A)(sin A− cos A)
Eyeballing and Mental Gymnastics
1. cosec2, sec2 suggest s2 + c2 ≡ 12. cosec = 1/sin, sec = 1/cos, cot = c/s, t = s/c3. rearrange and simplify.
LHS =sec A
cosec2 A−
cosec Asec2 A
=1
cos A·
(
sin2 A1
)
−
1sin A
(
cos2 A1
)
=sin2 Acos A
−
cos2 Asin A
=sin3 A− cos3 A∗
sin A cos A
=(sin A− cos A)(sin2 A+ sin A cos A+ cos2 A)
sin A cos A
= (sin A− cos A)(tan A+1+ cot A)
= (1+ cot A+ tan A)(sin A− cos A)
≡ RHS.
∗(a3 −b3) = (a−b)(a2 +ab+b2).
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
244 Trig or Treat
1+ sec Asec A
≡
sin2 A1− cos A
Eyeballing and Mental Gymnastics
1. sec = 1/cos2. s2 suggests s2 + c2 = 13. rearrange and simplify.
LHS =1+ sec A
sec A
=1+
1cos A1
cos A
=1+ cos A
cos A·
cos A1
= 1+ cos A
= (1+ cos A)
(
1− cos A1− cos A
)
=1− cos2 A1− cos A
=sin2 A
1− cos A
≡ RHS.
September 24, 2007 22:9 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 245
tan A− cot Atan A + cot A
≡ 1−2 cos2 A
Eyeballing and Mental Gymnastics
1. t = s/c, cot = c/s2. common denominators3. s2 + c2 ≡ 14. rearrange and simplify.
LHS =tan A− cot Atan A + cot A
=
sin Acos A
−
cos Asin A
sin Acos A
+cos Asin A
=
sin2 A− cos2 Acos A sin A
sin2 A + cos2 Acos A sin A
=sin2 A− cos2 A
sin2 A + cos2 A
= sin2 A− cos2 A
= (1− cos2 A)− cos2 A
= 1−2 cos2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
246 Trig or Treat
(2 cos2 A−1)2
cos4 A− sin4 A≡ 1−2 sin2 A
Eyeballing and Mental Gymnastics
1. c4 − s4 suggests (c2 − s2)(c2 + s2)2. s2 + c2 ≡ 13. rearrange and simplify.
LHS =(2 cos2 A−1)2
cos4 A− sin4 A
=(cos2 A− sin2 A)2
(cos2 A− sin2 A)(cos2 A+ sin2 A)
= cos2 A− sin2 A
= (1− sin2 A)− sin2 A
= 1−2 sin2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 247
1− tan2 A1+ tan2 A
+1 ≡ 2 cos2 A
Eyeballing and Mental Gymnastics
1. t = s/c2. s2 + c2 ≡ 13. rearrange and simplify.
LHS =1− tan2 A1+ tan2 A
+1
=1−
sin2 Acos2 A
1+sin2 Acos2 A
+1
=
cos2 A− sin2 Acos2 A
cos2 A+ sin2 Acos2 A
+1
=cos2 A− sin2 A
cos2 A+ sin2 1+1
= cos2 A− sin2 A+1
= cos2 A+ cos2 A
= 2 cos2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
248 Trig or Treat
1− sin A1+ sin A
≡ (sec A− tan A)2
Eyeballing and Mental Gymnastics
1. sec = 1/cos, t = s/c2. the RHS appears to be more complex and hence should be expanded first3. s2 + c2 ≡ 14. rearrange and simplify.
RHS = (sec A− tan A)2
=
(
1cos A
−
sin Acos A
)2
=
(
1− sin Acos A
)2
=(1− sin A)2
cos2 A
=(1− sin A)2
1− sin2 A
=(1− sin A)2
(1− sin A)(1+ sin A)
=1− sin A1+ sin A
≡ LHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips
Level-One-Games 249
(sec A− tan A)2 +1cosec A(sec A− tan A)
≡ 2 tan A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, t = s/c, cosec = 1/sin2. ( )2 suggests expansion3. s2 + c2 ≡ 14. rearrange and simplify.
LHS =(sec A− tan A)2 +1
cosec A(sec A− tan A)
=sec2 A−2 sec A tan A+ tan2 A+1
cosec A(sec A− tan A)
=sec2 A−2 sec A tan A+ sec2 A
cosec A(sec A− tan A)
=2 sec2 A−2 sec A tan Acosec A(sec A− tan A)
=2 sec A(sec A− tan A)
cosec A(sec A− tan A)
=2 sec Acosec A
= 21
cos A·
sin A1
= 2 tan A
≡ RHS.
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September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-GamesLess-Easy Proofs
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
252 Trig or Treat
2cot A tan2A
≡ 1− tan2 A
Eyeballing and Mental Gymnastics
1. 2A suggests expansion of “double angle”2. cot = 1/ tan3. rearrange and simplify.
LHS =2
cot A tan 2A
= 2 tan A ·
(1− tan2 A)
2 tan A
= 1− tan2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 253
cos4 A− sin4 A ≡
1sec 2A
Eyeballing and Mental Gymnastics
1. (cos4−sin4) suggests (a2 −b2)(a2 +b2)2. sec = 1/cos3. 2A suggests “double angle”4. rearrange and simplify.
LHS = cos4 A− sin4 A
= (cos2 A− sin2 A)(cos2 A+ sin2 A)
= (cos 2A)(1)
=1
sec 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
254 Trig or Treat
sin(A−B)
sin A cos B≡ 1− cot A tan B
Eyeballing and Mental Gymnastics
1. sin(A−B) suggests expansion2. cot = c/s, t = s/c3. rearrange and simplify.
LHS =sin(A−B)
sin A cos B
=sin A cos B− cos A sin B
sin A cos B
= 1− cot A tan B
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 255
cos(A−B)
cos A cos B≡ 1+ tan A tan B
Eyeballing and Mental Gymnastics
1. cos(A−B) suggests expansion2. t = s/c3. rearrange and simplify.
LHS =cos(A−B)
cos A cos B
=cos A cos B+ sin A sin B
cos A cos B
= 1+ tan A tan B
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
256 Trig or Treat
cos(A+B)
sin A cos B≡ cot A− tan B
Eyeballing and Mental Gymnastics
1. cos(A+B) suggests expansion2. cot = c/s, t = s/c3. rearrange and simplify.
LHS =cos(A+B)
sin A cos B
=cos A cos B− sin A sin B
sin A cos B
= cot A− tan B
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 257
cos(A−B)
sin A cos B≡ cot A+ tan B
Eyeballing and Mental Gymnastics
1. cos(A−B) suggests expansion2. cot = c/s, t = s/c3. rearrange and simplify.
LHS =cos(A−B)
sin A cos B
=cos A cos B+ sin A sin B
sin A cos B
= cot A+ tan B
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
258 Trig or Treat
cos(A−B)
sin A sin B≡ 1+ cot A cot B
Eyeballing and Mental Gymnastics
1. cos(A−B) suggests expansion2. cot = c/s3. rearrange and simplify.
LHS =cos(A−B)
sin A sin B
=cos A cos B+ sin A sin B
sin A sin B
=cos A cos Bsin A sin B
+1
= 1+ cot Acot B
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 259
sin(A+B)
cos A cos B≡ tan A+ tan B
Eyeballing and Mental Gymnastics
1. sin(A+B) suggests expansion2. t = s/c3. rearrange and simplify.
LHS =sin(A+B)
cos A cos B
=sin A cos B+ cos A sin B
cos A cos B
=sin Acos A
+sin Bcos B
= tan A+ tan B
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
260 Trig or Treat
cos4 A− sin4 A ≡ cos 2A
Eyeballing and Mental Gymnastics
1. c4 − s4 suggests (a2 −b2)(a2 +b2)2. s2 + c2 ≡ 13. rearrange and simplify.
LHS = cos4 A− sin4 A
= (cos2 A− sin2 A)(cos2 A+ sin2 A)
= (cos 2A)(1)
= cos 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 261
(2a sin A cos A)2 +a2(cos2 A− sin2 A)2≡ a2
Eyeballing and Mental Gymnastics
1. 2 sin A cos A suggests sin 2A2. cos2 A− sin2 A suggests cos 2A3. s2 + c2 ≡ 14. rearrange and simplify.
LHS = (2a sin A cos A)2 +a2(cos2 A− sin2 A)2
= a2(sin 2A)2 +a2(cos 2A)2
= a2(sin2 2A+ cos2 2A)
= a2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
262 Trig or Treat
sin A1+ cos A
≡ tanA2
Eyeballing and Mental Gymnastics
1. A/2 suggests use of “half angle formulas” for sin A, cos A2. t = s/c3. rearrange and simplify.
LHS =sin A
1+ cos A
=2 sin
A2
cosA2
1+
(
2 cos2 A2−1
)
=2 sin
A2
cosA2
2 cos2 A2
= tanA2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 263
1− cos Asin A
≡ tanA2
Eyeballing and Mental Gymnastics
1. A/2 suggests “half angle formulas” for cos A, sin A2. rearrange and simplify.
LHS =1− cos A
sin A
=
1−
(
1−2 sin2 A2
)
2 sinA2
cosA2
=2 sin2 A
2
2 sinA2
cosA2
=sin
A2
cosA2
= tanA2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
264 Trig or Treat
(
sinA2
+ cosA2
)2
≡ 1+ sin A
Eyeballing and Mental Gymnastics
1. A/2 suggests “half angle formulas”2. ( )2 suggests s2 + c2 ≡ 13. rearrange and simplify.
LHS =
(
sinA2
+ cosA2
)2
= sin2 A2
+2 sinA2
cosA2
+ cos2 A2
= sin2 A2
+ cos2 A2
+2 sinA2
cosA2
= 1+ sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 265
cos(60◦ +A)+ sin(30◦ +A) ≡ cos A
Eyeballing and Mental Gymnastics
1. (60◦+A), (30◦ +A) suggest expansion of “compound angles”2. numerical values for sin, cos of 60◦, 30◦
3. rearrange and simplify.
LHS = cos(60◦ +A)+ sin(30◦ +A)
= cos 60◦ cos A− sin 60◦ sin A
+ sin 30◦ cos A+ cos 30◦ sin A
=12
cos A−
√
32
sin A+12
cos A+
√
32
sin A
= cos A
≡ RHS.
sin cos
0◦ 0 =
√
04
√
44
= 1
30◦12
=
√
14
√
34
=
√
32
45◦√
22
=
√
24
√
24
=
√
22
60◦√
32
=
√
34
√
14
=12
90◦ 1 =
√
44
√
04
= 0
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
266 Trig or Treat
1+ sin 2A+ cos 2Asin A+ cos A
≡ 2 cos A
Eyeballing and Mental Gymnastics
1. 2A suggests expansion of “double angle”2. rearrange and simplify.
LHS =1+ sin 2A+ cos 2A
sin A+ cos A
=1+2sin A cos A+(2 cos2 A−1)
sin A+ cos A
2cos A(sin A+ cos A)
sin A+ cos A
= 2 cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 267
sin 2A1− cos 2A
≡ cot A
Eyeballing and Mental Gymnastics
1. 2A suggests expansion of “double angle”2. s2 + c2 ≡ 1, cot = c/s3. rearrange and simplify.
LHS =sin 2A
1− cos 2A
=2 sin A cos A
1− (1−2 sin2 A)
=2 sin A cos A
2 sin2 A
=cos Asin A
= cot A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
268 Trig or Treat
sin A+ sin 2A1+ cos A+ cos 2A
≡ tan A
Eyeballing and Mental Gymnastics
1. 2A suggests expansion of “double angle”2. t = s/c3. rearrange and simplify.
LHS =sin A+ sin 2A
1+ cos A+ cos 2A
=sin A+2sin A cos A
1+ cos A+(2 cos2 A−1)
=sin A(1+2 cos A)
cos A+2 cos2 A
=sin A(1+2 cos A)
cos A(1+2 cos A)
= tan A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 269
12(cot A− tan A) ≡ cot 2A
Eyeballing and Mental Gymnastics
1. cot = c/s, t = s/c2. cot 2A = cos 2A/sin 2A3. rearrange and simplify.
LHS =12(cot A− tan A)
=12
(
cos Asin A
−
sin Acos A
)
=12
(
cos2−sin2 A
sin A cos A
)
=cos 2Asin 2A
= cot 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
270 Trig or Treat
cosec A sec A ≡ 2 cosec 2A
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, sec = 1/cos2. cosec 2A suggests 1/sin 2A3. rearrange and simplify.
LHS = cosec A sec A
=1
sin A·
1cos A
=1
sin A cos A·
(
22
)
=2
2sin A cos A
=2
sin 2A
= 2 cosec 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 271
cos(A+B)
cos Asin B≡ cot B− tan A
Eyeballing and Mental Gymnastics
1. cos(A+B) suggests expansion2. cot = c/s3. rearrange and simplify.
LHS =cos(A+B)
cos A sin B
=cos A cos B− sin A sin B
cos A sin B
= cot B− tan A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
272 Trig or Treat
sec A sec B1+ tan A tan B
≡ sec(A−B)
Eyeballing and Mental Gymnastics
1. sec(A−B) suggests expansion2. sec = 1/cos, t = s/c3. begin with RHS since sec(A − B) = 1/cos(A − B) which is standard
“compound angle” function4. rearrange and simplify.
RHS = sec(A−B)
=1
cos(A−B)
=1
cos A cos B+ sin A sin B
=sec A sec B
1+ tan A tan B
≡ LHS.
divide bothnumeratorand denominatorby cos A cos B
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 273
cosec Acosec Bcot A cot B−1
≡ sec(A+B)
Eyeballing and Mental Gymnastics
1. sec(A+B) suggests expansion2. cosec = 1/sin, sec = 1/cos, cot = c/s3. begin with RHS since sec(A + B) = 1/cos(A + B) which is a standard
“compound angle” function.4. rearrange and simplify.
RHS = sec(A+B)
=1
cos(A+B)
=1
cos A cos B− sin A sin B
=cosec Acosec Bcot A cot B−1
≡ LHS.
divide bothnumeratorand denominatorby sin A sin B
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
274 Trig or Treat
cot A cot B+1cot B− cot A
≡ cot(A−B)
Eyeballing and Mental Gymnastics
1. cot(A−B) suggests expansion2. cot = 1/ tan3. begin with RHS since cot(A−B) = 1/ tan(A−B), and tan(A−B) is a standard
“compound angle” function4. rearrange and simplify.
RHS = cot(A−B)
=1
tan(A−B)
=1+ tan A tan Btan A− tan B
=cot A cot B+1cot B− cot A
≡ LHS.
divide bothnumeratorand denominatorby tan A tan B
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 275
cos(A+B)
cos A cos B≡ 1− tan A tan B
Eyeballing and Mental Gymnastics
1. cos(A+B) suggests expansion2. t = s/c3. rearrange and simplify.
LHS =cos(A+B)
cos A cos B
=cos A cos B− sin A sin B
cos A cos B
= 1−sin A sin Bcos A cos B
= 1− tan A tan B
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
276 Trig or Treat
sin(A+B)
sin A cos B≡ 1+ cot A tan B
Eyeballing and Mental Gymnastics
1. sin(A+B) suggests expansion2. cot = c/s, t = s/c3. rearrange and simplify.
LHS =sin(A+B)
sin A cos B
=sin A cos B+ cos A sin B
sin A cos B
= 1+cos A sin Bsin A cos B
= 1+ cot A tan B
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 277
1− tan A tan B1+ tan A tan B
≡
cos(A+B)
cos(A−B)
Eyeballing and Mental Gymnastics
1. cos(A+B), cos(A−B) suggest expansion2. t = s/c3. begin with RHS which are standard “compound angle” functions4. rearrange and simplify.
RHS =cos(A+B)
cos(A−B)
=cos A cos B− sin A sin Bcos A cos B+ sin A sin B
=1− tan A tan B1+ tan A tan B
≡ LHS.
divide bothnumeratorand denominatorby cos A cos B
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
278 Trig or Treat
tan A+ tan Btan A− tan B
≡
sin(A+B)
sin(A−B)
Eyeballing and Mental Gymnastics
1. sin(A+B), sin(A−B) suggest expansion2. t = s/c3. begin with RHS which are standard “compound angle” functions4. rearrange and simplify.
RHS =sin(A+B)
sin(A−B)
=sin A cos B+ cos A sin Bsin A cos B− cos A sin B
=tan A+ tan Btan A− tan B
≡ LHS.
divide bothnumeratorand denominatorby cos A cos B
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 279
cos A− sin Acos A+ sin A
+cos A+ sin Acos A− sin A
≡ 2 sec 2A
Eyeballing and Mental Gymnastics
1. sec = 1/cos2. 2A suggests “double angle”3. common denominator4. rearrange and simplify.
LHS =cos A− sin Acos A+ sin A
+cos A+ sin Acos A− sin A
=(cos A− sin A)2 +(cos A+ sin A)2
(cos A+ sin A)(cos A− sin A)
=(cos2 A−2 sin A cos A+ sin2 A)+(cos2 A+2 sin A cos A+ sin2 A)
cos2 A− sin2 A
=(1)+(1)
cos 2A
= 2 sec 2A
≡RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
280 Trig or Treat
cot A− tan Acot A+ tan A
≡ cos 2A
Eyeballing and Mental Gymnastics
1. cot = c/s, t = s/c2. common denominators3. rearrange and simplify.
LHS =cot A− tan Acot A+ tan A
=
cos Asin A
−
sin Acos A
cos Asin A
+sin Acos A
=cos2 A− sin2 A
sin A cos A·
sin A cos A
cos2 A+ sin2 A
=cos2 A− sin2 A
cos2 A+ sin2 A
=cos 2A
1
= cos 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 281
cos A− cos Bsin A+ sin B
+sin A− sin Bcos A+ cos B
≡ 0
Eyeballing and Mental Gymnastics
1. common denominators2. rearrange and simplify.
LHS =
(
cos A− cos Bsin A+ sin B
)
+
(
sin A− sin Bcos A+ cos B
)
=(cos A− cos B)(cos A+ cos B)+(sin A− sin B)(sin A+ sin B)
(sin A+ sin B)(cos A+ cos B)
=(cos2 A− cos2 B)+(sin2 A− sin2 B)
(sin A+ sin B)(cos A+ cos B)
=(cos2 A+ sin2 A)− (cos2 B+ sin2 B)
(sin A+ sin B)(cos A+ cos B)
=(1)− (1)
(sin A+ sin B)(cos A+ cos B)
= 0
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
282 Trig or Treat
cos A+ sin Acos A− sin A
−
cos A− sin Acos A+ sin A
≡ 2 tan 2A
Eyeballing and Mental Gymnastics
1. common denominators2. tan 2A suggest “double angle” formula3. lots of cos2, sin2 suggest s2 + c2 ≡ 14. rearrange and simplify.
LHS =
(
cos A+ sin Acos A− sin A
)
−
(
cos A− sin Acos A+ sin A
)
=(cos A+ sin A)(cos A+ sin A)− (cos A− sin A)(cos A− sin A)
(cos A− sin A)(cos A+ sin A)
=(cos2 A+2 sin A cos A+ sin2 A)− (cos2 A−2 sin A cos A+ sin2 A)
cos2 A− sin2 A
=4 sin A cos A
cos 2A
= 2sin 2Acos 2A
= 2 tan 2A
≡RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 283
(4 sin A cos A)(1−2 sin2 A) ≡ sin 4A
Eyeballing and Mental Gymnastics
1. 4 sin A cos A = 2(2 sin A cos A) = 2 sin 2A2. (1−2 sin2 A) = cos 2A3. rearrange and simplify.
LHS = 4 sin A cos A(1−2 sin2 A)
= 2(2 sin A cos A)(cos 2A)
= 2(sin 2A)(cos 2A)
= sin 4A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
284 Trig or Treat
sin 3A cos A− sin A cos 3Asin 2A
≡ 1
Eyeballing and Mental Gymnastics
1. sin A cos B− cos A sin B = sin(A−B)2. rearrange and simplify.
LHS =sin 3A cos A− sin A cos 3A
sin 2A
=sin(3A−A)
sin 2A
=sin 2Asin 2A
= 1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 285
tan(
A−
π4
)
≡
tan A−1tan A+1
Eyeballing and Mental Gymnastics
1. t = s/c2. (A− (π/4)) suggests expansion of “compound angle”3. rearrange and simplify.
LHS = tan(
A−
π4
)
=tan A− tan
π4
1+ tan A tanπ4
=tan A−11+ tan A
=tan A−1tan A+1
≡ RHS.
tanπ4
= 1
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
286 Trig or Treat
cos3 A− sin3 Acos A− sin A
≡
2+ sin 2A2
Eyeballing and Mental Gymnastics
1. cos3−sin3 suggests (a3 −b3) = (a−b)(a2 +ab+b2)2. sin 2A suggests “double angle”3. rearrange and simplify.
LHS =cos3 A− sin3 Acos A− sin A
=(cos A− sin A)(cos2 A+ cos A sin A+ sin2 A)
(cos A− sin A)
= (1+ cos A sin A)
=12(2+2 cos A sin A)
=2+ sin 2A
2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 287
sin 3Asin A
−
cos 3Acos A
≡ 2
Eyeballing and Mental Gymnastics
1. common denominator2. sin(A−B) expansion3. rearrange and simplify.
LHS =sin 3Asin A
−
cos 3Acos A
=sin 3A cos A− cos 3A sin A
sin A cos A
=sin(3A−A)
sin A cos A
=sin 2A
12(2 sin A cos A)
= 2sin 2Asin 2A
= 2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
288 Trig or Treat
2 cot A cot 2A ≡ cot2 A−1
Eyeballing and Mental Gymnastics
1. cot = c/s2. cos 2A, sin 2A suggest expansion3. rearrange and simplify.
LHS = 2 cot A cot 2A
= 2cos Asin A
·
cos 2Asin 2A
= 2cos Asin A
·
(cos2 A− sin2 A)
2 sin A cos A
=1
sin2 A(cos2 A− sin2 A)
= cot2 A−1
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 289
12
sec Acosec A ≡ cosec 2A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, cosec = 1/sin2. cosec 2A = 1/sin 2A3. rearrange and simplify.
LHS =12
sec Acosec A
=12
1cos A
·
1sin A
=1
2 sin A cos A
=1
sin 2A
= cosec 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
290 Trig or Treat
tan A+ cot A ≡
2sin 2A
Eyeballing and Mental Gymnastics
1. t = s/c, cot = c/s2. sin 2A suggests “double angle”3. rearrange and simplify.
LHS = tan A+ cot A
=sin Acos A
+cos Asin A
=sin2 A+ cos2 A
sin A cos A
=1
sin A cos A
=2
2 sin A cos A
=2
sin 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 291
21− cos A
≡ cosec 2 A2
Eyeballing and Mental Gymnastics
1. A/2 suggests expansion of cos A2. cosec = 1/sin3. rearrange and simplify.
LHS =2
1− cos A
=2
1−
(
1−2 sin2 A2
)
=2
2 sin2 A2
=1
sin2 A2
= cosec 2 A2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
292 Trig or Treat
21+ cos A
≡ sec2 A2
Eyeballing and Mental Gymnastics
1. A/2 suggests expansion of cos A2. sec2 = 1/cos2
3. rearrange and simplify.
LHS =2
1+ cos A
=2
1+
(
2 cos2 A2−1
)
=2
2 cos2 A2
=1
cos2 A2
= sec2 A2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 293
(1+ cos A) tanA2≡ sin A
Eyeballing and Mental Gymnastics
1. tan(A/2) suggests expansion of cos A as cos 2(A/2) and sin A as sin 2(A/2)2. rearrange and simplify.
LHS = (1+ cos A) tanA2
=
(
1+ cos 2
(
A2
))
tanA2
=
(
1+
(
2 cos2 A2−1
))
tanA2
= 2 cos2 A2·
sinA2
cosA2
= 2 cosA2
sinA2
= sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
294 Trig or Treat
sin3 A+ cos3 Asin A+ cos A
≡ 1−sin 2A
2
Eyeballing and Mental Gymnastics
1. s3 + c3 = (s+ c)(s2 − sc+ c2)2. s2 + c2 ≡ 13. sin 2A = 2 sin A cos A4. rearrange and simplify.
LHS =sin3 A+ cos3 Asin A+ cos A
=(sin A+ cos A)(sin2 A− sin A cos A+ cos2 A)
(sin A+ cos A)
= sin2 A− sin A cos A+ cos2 A
= 1− sin A cos A
= 1−12(2 sin A cos A)
= 1−sin 2A
2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 295
sec2 A2− sec2 A
≡ sec 2A
Eyeballing and Mental Gymnastics
1. sec = 1/cos, sec 2A = 1/cos 2A2. sec2 suggests s2 + c2 ≡ 13. rearrange and simplify.
LHS =sec2 A
2− sec2 A
=1
cos2 A·
1
2−1
cos2 A
=1
cos2 A·
cos2 A(2 cos2 A−1)
=1
2 cos2 A−1
=1
cos 2A
= sec 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
296 Trig or Treat
cosec A− cot A ≡ tanA2
Eyeballing and Mental Gymnastics
1. cosec = 1/sin, cot = c/s, t = s/c2. A/2 suggests expansion of sin A, cos A3. rearrange and simplify.
LHS = cosec A− cot A
=1
sin A−
cos Asin A
=1− cos A
sin A
=
1−
(
1−2 sin2 A2
)
2 sinA2
cosA2
=2 sin2 A
2
2 sinA2
cosA2
=sin
A2
cosA2
= tanA2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 297
sin A tanA2≡ 1− cos A
Eyeballing and Mental Gymnastics
1. tan(A/2) suggests expansion of sin A as sin 2(A/2), and cos A as cos 2(A/2)2. t = s/c3. rearrange and simplify.
LHS = sin A tanA2
= sin 2
(
A2
)
tanA2
= 2 sinA2· cos
A2·
sinA2
cosA2
= 2 sin2 A2
=
(
2 sin2 A2−1
)
+1
= 1− cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
298 Trig or Treat
sin 2A tan A ≡ 1− cos 2A
Eyeballing and Mental Gymnastics
1. Expand “double angle”2. t = s/c3. rearrange and simplify.
LHS = sin 2A tan A
= 2 sin A cos A ·
sin Acos A
= 2 sin2 A
= (2 sin2 A−1)+1
= 1− cos 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 299
cosec 2A− cot 2A ≡ tan A
Eyeballing and Mental Gymnastics
1. 2A suggests expansion of “double angle”2. cosec = 1/sin, cot = c/s, t = s/c3. rearrange and simplify.
LHS = cosec 2A− cot 2A
=1
sin 2A−
cos 2Asin 2A
=1− cos 2A
sin 2A
=1− (1−2 sin2 A)
2 sin A cos A
=sin Acos A
= tan A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
300 Trig or Treat
tan
(
45◦ +A2
)
≡ tan A+ sec A
Eyeballing and Mental Gymnastics
1. (45◦+A/2) suggests expansion of “compound angle”2. A/2 suggests “half angle” expansion3. both sides are complex; hence explore simplification on both sides to achieve
identity.
LHS = tan
(
45◦ +A2
)
=tan 45◦ + tan
A2
1− tan 45◦ tanA2
=1+ tan
A2
1− tanA2
RHS = tan A+ sec A
=sin Acos A
+1
cos A
=sin A+1
cos A
=
2 sinA2
cosA2
+
(
cos2 A2
+ sin2 A2
)
cos2 A2− sin2 A
2
=
(
sinA2
+ cosA2
)2
(
cosA2
+ sinA2
)(
cosA2− sin
A2
) =
(
sinA2
+ cosA2
)
(
cosA2− sin
A2
)
=tan
A2
+1
1− tanA2
≡ simplified form of LHS.
divide numeratorand denominator
by cosA2
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 301
A more elegant proof comes from the use of the application of Pythago-ras Theorem with respect to the “half angle formula” of tan A.
tan A =2 tan
A2
1− tan2 A2
.
Let t = tan(A/2).Then from Pythagoras Theorem
the hypotenuse =√
(2t)2 +(1− t2)2
=√
1+2(2t)+ t4
= 1+ t2
∴ cos A =1− t2
1+ t2
∴ RHS = tan A+ sec A
= tan A+1
cos A
=2t
1− t2 +1+ t2
1− t2 where t = tanA2
=1+2t + t2
(1− t2)
=(1+ t)2
(1+ t)(1− t)
=1+ t1− t
=1+ tan
A2
1− tanA2
≡ simplified form of LHS.
1 − t2
1 + t2
2t
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
302 Trig or Treat
tan Atan 2A− tan A
≡ cos 2A
Eyeballing and Mental Gymnastics
1. t = s/c2. 2A suggests expansion of “compound angle”3. rearrange and simplify.
LHS =tan A
tan 2A− tan A
=sin Acos A
·
1(
sin 2Acos 2A
−
sin Acos A
)
=sin Acos A
·
cos Acos 2A(sin 2A cos A− sin A cos 2A)
=sin A cos 2A
(sin 2A cos A− cos 2A sin A)
=sin A cos 2Asin(2A−A)
= cos 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 303
sec 2A− tan 2A ≡ tan(45◦−A)
Eyeballing and Mental Gymnastics
1. 2A, (45◦−A) suggest expansion of “compound angles”2. since both LHS and RHS have complex functions, explore simplification of
both side to achieve identity.3. sec = 1/cos, t = s/c4. rearrange and simplify.
LHS = sec 2A− tan 2A
=1
cos 2A−
sin 2Acos 2A
=1− sin 2A
cos 2A
=(cos2 A+ sin2 A−2 sin A cos A)
cos2 A− sin2 A
=(cos A− sin A)2
(cos A+ sin A)(cos A− sin A)
=(cos A− sin A)
(cos A+ sin A)
=1− tan A1+ tan A
RHS = tan(45◦−A)
=tan 45◦− tan A
1+ tan 45◦ tan A
=1− tan A1+ tan A
≡ LHS.
dividing all termsby cos A to preparefor comparisonwith RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
304 Trig or Treat
2 sin2 A6− sin2 A
7≡ cos2 A
7− cos
A3
Eyeballing and Mental Gymnastics
1. sin2(A/7), cos2(A/7) suggest s2 + c2 ≡ 12. A/3 suggests “double angle” formula to give A/63. rearrange and simplify.
LHS = 2 sin2 A6− sin2 A
7
= 2 sin2 A6−
(
1− cos2 A7
)
=
(
2 sin2 A6−1
)
+ cos2 A7
= −cos 2
(
A6
)
+ cos2 A7
= cos2 A7− cos
A3
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 305
sin 2A1+ cos 2A
≡ tan A
Eyeballing and Mental Gymnastics
1. 2A suggests expansion of “double angle”2. rearrange and simplify.
LHS =sin 2A
1+ cos 2A
=2 sin A cos A
1+(2 cos2 A−1)
=2 sin A cos A
2 cos2 A
=sin Acos A
= tan A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
306 Trig or Treat
tan A+ cot A ≡ 2cosec 2A
Eyeballing and Mental Gymnastics
1. t = s/c, cot = c/s, cosec = 1/sin2. 2A suggests “double angle”3. rearrange and simplify.
LHS = tan A+ cot A
=sin Acos A
+cos Asin A
=sin2 A+ cos2 A
cos A sin A
=1
cos A sin A
=22·
1cos A sin A
=2
sin 2A
= 2cosec 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 307
2 cos2(45◦−A) ≡ 1+ sin 2A
Eyeballing and Mental Gymnastics
1. c2 suggests s2 + c2 ≡ 12. (45◦−A), 2A suggest expansion of “compound angles”3. rearrange and simplify.
LHS = 2 cos2(45◦−A)
= 2(cos 45◦ cos A+ sin 45◦ sin A)2
= 2
(√
22
cos A+
√
22
sin A
)2
= 2
(√
22
)2
(cos A+ sin A)2
= cos2 A+2 sin A cos A+ sin2 A
= 1+ sin 2A
≡ RHS.
sin cos
45◦√
24
√
24
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
308 Trig or Treat
cos(A+B)cos(A−B) ≡ cos2 B− sin2 A
Eyeballing and Mental Gymnastics
1. Expand “compound angles”2. c2, s2 suggest s2 + c2 ≡ 13. rearrange and simplify.
LHS = cos(A+B)cos(A−B)
= (cos A cos B− sin A sin B)(cos A cos B+ sin A sin B)
= cos2 A cos2 B− sin2 A sin2 B
= cos2 B(1− sin2 A)− sin2 A sin2 B
= cos2 B− cos2 B sin2 A− sin2 A sin2 B
= cos2 B− sin2 A(cos2 B+ sin2 B)
= cos2 B− sin2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 309
cos(A+B)cos(A−B) ≡ cos2 A− sin2 B
Eyeballing and Mental Gymnastics
1. Expand “compound angles”2. c2, s2 suggest s2 + c2 ≡ 13. rearrange and simplify.
LHS = cos(A+B)cos(A−B)
= (cos A cos B− sin A sin B)(cos A cos B+ sin A sin B)
= cos2 A cos2 B− sin2 A sin2 B
= cos2 A(1− sin2 B)− sin2 A sin2 B
= cos2 A− cos2 A sin2 B− sin2 A sin2 B
= cos2 A− sin2 B(cos2 A+ sin2 A)
= cos2 A− sin2 B
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
310 Trig or Treat
sin(A+B)sin(A−B)≡ sin2 A− sin2 B
Eyeballing and Mental Gymnastics
1. Expand “compound angles”2. s2 suggests s2 + c2 ≡ 13. rearrange and simplify.
LHS = sin(A+B)sin(A−B)
= (sin A cos B+ cos A sin B)(sin A cos B− cos A sin B)
= sin2 A cos2 B− cos2 A sin2 B
= sin2 A(1− sin2 B)− cos2 A sin2 B
= sin2 A− sin2 A sin2 B− cos2 A sin2 B
= sin2 A− sin2 B(sin2 A+ cos2 A)
= sin2 A− sin2 B
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 311
sin(A+B)sin(A−B) ≡ cos2 B− cos2 A
Eyeballing and Mental Gymnastics
1. Expand “compound angles”2. c2 suggests s2 + c2 ≡ 13. rearrange and simplify.
LHS = sin(A+B)sin(A−B)
= (sin A cos B+ cos A sin B)(sin A cos B− cos A sin B)
= sin2 A cos2 B− cos2 A sin2 B
= cos2 B(1− cos2 A)− cos2 A sin2 B
= cos2 B− cos2 A cos2 B− cos2 A sin2 B
= cos2 B− cos2 A(cos2 B+ sin2 B)
= cos2 B− cos2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
312 Trig or Treat
sec A+1sec A−1
≡ cot2A2
Eyeballing and Mental Gymnastics
1. sec = 1/cos, cot = c/s2. A/2 suggests expansion of cos A3. cot2 suggests s2 + c2 ≡ 14. rearrange and simplify.
LHS =sec A+1sec A−1
=
(
1cos A
+1
)
(
1cos A
−1
)
=
(
1+ cos Acos A
)
·
(
cos A1− cos A
)
=1+ cos A1− cos A
=
1+
(
2 cos2 A2−1
)
1−
(
1−2 sin2 A2
)
=2 cos2 A
2
2 sin2 A2
= cot2A2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 313
sec2 A2− sec2 A
≡ sec 2A
Eyeballing and Mental Gymnastics
1. sec = 1/cos2. 2A suggests “compound angle”3. rearrange and simplify.
LHS =sec2 A
2− sec2 A
=1
cos2 A·
1(
2−1
cos2 A
)
=1
cos2 A·
cos2 A(2 cos2 A−1)
=1
cos 2A
= sec2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
314 Trig or Treat
18(1− cos 4A) ≡ sin2 A cos2 A
Eyeballing and Mental Gymnastics
1. cos 4A suggests expansion twice: cos 4A → cos 2A → cos A2. s2, c2 suggest s2 + c2 ≡ 13. rearrange and simplify.
LHS =18(1− cos 4A)
=18(1− (2 cos2 2A−1))
=18(2−2(2 cos2 A−1)2)
=18·2(1− (4 cos4 A−4 cos2 A+1))
=14(4 cos2 A(cos2 A−1))
= cos2 A(sin2 A)
= sin2 A cos2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 315
cot2 A−12 cot A
≡ cot 2A
Eyeballing and Mental Gymnastics
1. cot = c/s, t = s/c2. cot 2A = cos 2A/sin 2A3. rearrange and simplify.
LHS =cot2 A−1
2 cot A
=12
(
cot A−
1cot A
)
=12(cot A− tan A)
=12
(
cos Asin A
−
sin Acos A
)
=12
(
cos2 A− sin2 Asin A cos A
)
=cos 2Asin 2A
= cot 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
316 Trig or Treat
2 tan A1+ tan2 A
≡ sin 2A
Eyeballing and Mental Gymnastics
1. t = s/c2. 2A suggests “double angle”3. rearrange and simplify.
LHS =2 tan A
1+ tan2 A
=
2
(
sin Acos A
)
1+
(
sin Acos A
)2
= 2
(
sin Acos A
)(
cos2 A
cos2 A+ sin2 A
)
= 2sin Acos A
(
cos2 A1
)
= 2 sin A cos A
= sin 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 317
1− tan2 A1+ tan2 A
≡ cos 2A
Eyeballing and Mental Gymnastics
1. t = s/c2. 2A suggests “double angle”3. rearrange and simplify.
LHS =1− tan2 A1+ tan2 A
=
(
1−
(
sin Acos A
)2)
1(
1+
(
sin Acos A
)2)
=
(
cos2 A− sin2 Acos2 A
)(
cos2 A
cos2 A+ sin2 A
)
=(cos2 A− sin2 A)
cos2 A·
cos2 A1
= cos2 A− sin2 A
= cos 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
318 Trig or Treat
The previous two identities are part of the series of tan 2A, sin 2A andcos 2A in terms of tan A.
tan 2A =2 tan A
1− tan2 A
sin 2A =2 tan A
1+ tan2 A
cos 2A =1− tan2 A1+ tan2 A
The “double angle” formula for tan is well known to generations ofstudents. But few are those who know about these special formulas forsin2A and cos2A in terms of tan A. You are among the very few. Test itout yourself with your friends!
An easy way to remember the three identities is to write tan A as t.
then: tan 2A =2t
1− t2
From Pythagoras Theorem, the hypotenuse is given by:
√
(2t)2 +(1− t2)2
=√
4t2 +1−2t2 + t4
=√
1+2t2 + t4
= 1+ t2
Then:
sin 2A =OH
=2t
1+ t2 , and
cos 2A =AH
=1− t2
1+ t2 .
1 − t2
1 + t2
2t
2A
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 319
This sin 2A equation is one of the amazing equations in Trigonometrywhere a slight difference (from minus to plus sign in the denominator)changes the tangent identity for the double angle to the sine identity for thesame double angle.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
320 Trig or Treat
cos 2Asin A
+sin 2Acos A
≡ cosec A
Eyeballing and Mental Gymnastics
1. 2A suggests expansion of “double angle”2. cosec = 1/sin3. rearrange and simplify.
LHS =cos 2Asin A
+sin 2Acos A
=cos2 A− sin2 A
sin A+
2 sin A cos Acos A
=1−2 sin2 A
sin A+2 sin A
=1−2 sin2 A+2 sin2 A
sin A
=1
sin A
= cosec A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 321
2 sin(A−B)
cos(A+B)− cos(A−B)≡ cot A− cot B
Eyeballing and Mental Gymnastics
1. (A+B), (A−B) suggest expansion of “compound angles”2. cot = c/s3. rearrange and simplify.
LHS =2 sin(A−B)
cos(A+B)− cos(A−B)
=2(sin A cos B− cos A sin B)
(cos A cos B− sin A sin B)− (cos A cos B+ sin A sin B)
=2(sin A cos B− cos A sin B)
−2 sin A sin B
=(cos A sin B− sin A cos B)
sin A sin B
= cot A− cot B
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
322 Trig or Treat
cos 2A1+ sin 2A
≡
cot A−1cot A+1
Eyeballing and Mental Gymnastics
1. cos 2A, sin 2A suggest expansion2. cot = c/s3. rearrange and simplify.
LHS =cos 2A
1+ sin 2A
=(cos2 A− sin2 A)
1+2 sin A cos A
=cos2 A− sin2 A
cos2 A+ sin2 A+2 sin A cos A
=(cos A− sin A)(cos A+ sin A)
(cos A+ sin A)2
=cos A− sin Acos A+ sin A
=cot A−1cot A+1
≡ RHS.
dividing both thenumerator andthe denominatorby sin.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 323
cot Acot B−1cot A+ cot B
≡ cot(A+B)
Eyeballing and Mental Gymnastics
1. (A+B) on RHS suggests expansion2. cot = 1/ tan3. begin with RHS expansion of tan(A+B)4. rearrange and simplify.
RHS = cot(A+B)
=1
tan(A+B)
=1− tan A tan Btan A+ tan B
=cot A cot B−1cot B+ cot A
=cot A cot B−1cot A+ cot B
≡ LHS.
divide bothnumeratorand denominatorby tan A tan B
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
324 Trig or Treat
3 sin A−4 sin3 A ≡ sin 3A
Eyeballing and Mental Gymnastics
1. 3A suggests expansion of “compound angle” twice2. rearrange and simplify3. the RHS is a standard function, easy to expand, twice.
RHS = sin 3A
= sin(2A+A)
= sin 2A cos A+ cos 2A sin A
= 2 sin A cos A · cos A+ sin A(cos2 A− sin2 A)
= 2 sin A cos2 A+ sin A cos2 A− sin3 A
= 3 sin A cos2 A− sin3 A
= 3 sin A(1− sin2 A)− sin3 A
= 3 sin A−3 sin3 A− sin3 A
= 3 sin A−4 sin3 A
≡ LHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 325
3 tan A− tan3 A1−3 tan2 A
≡ tan 3A
Eyeballing and Mental Gymnastics
1. RHS tan 3A is standard double expansion of tan(2A+A) and tan 2A.2. begin with RHS (an exception to normal practice)3. rearrange and simplify.
RHS = tan 3A
= tan(2A+A)
=tan 2A+ tan A
1− tan 2A tan A
=
2 tan A1− tan2 A
+ tan A
1−2 tan A
1− tan2 Atan A
=
2 tan A+ tan A− tan3 A1− tan2 A
1− tan2 A−2tan2 A1− tan2 A
=3 tan A− tan3 A
1−3 tan2 A
≡ LHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
326 Trig or Treat
tan(45◦ +A) tan(45◦−A) ≡ cot(45◦ +A)cot(45◦−A)
Eyeballing and Mental Gymnastics
1. tan 45◦ = 1, cot 45◦ = 12. both sides equally complex; therefore maybe easier to work on both sides to
reduce to common terms3. ( ) suggests expansion4. rearrange and simplify.
LHS = tan(45◦ +A) tan(45◦−A)
=(tan 45◦ + tan A)
(1− tan 45◦ tan A)·
(tan 45◦− tan A)
(1+ tan 45◦ tan A)
=(1+ tan A)
(1− tan A)·
(1− tan A)
(1+ tan A)
= 1
RHS = cot(45◦ +A)cot(45◦−A)
=1
tan(45◦ +A)·
1tan(45◦−A)
=(1− tan 45◦ tan A)
(tan 45◦ + tan A)·
(1+ tan 45◦ tan A)
(tan 45◦− tan A)
=(1− tan A)
(1+ tan A)·
(1+ tan A)
(1− tan A)
= 1
≡ simplified form of LHS.
tan 45◦ = 1cot 45◦ = 1
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 327
sin A+ sin 2A2+3 cos A+ cos 2A
≡ tanA2
Eyeballing and Mental Gymnastics
1. sin 2A, cos 2A suggest “double angle” expansion2. (A/2) on RHS suggests “half-angle formula”3. rearrange and simplify.
LHS =sin A+ sin 2A
2+3 cos A+ cos 2A
=sin A+2 sin A cos A
2+3 cos A+(2 cos2 A−1)
=sin A(1+2 cos A)
1+3 cos A+2 cos2 A
=sin A(1+2 cos A)
(1+ cos A)(1+2 cos A)
=2 sin
A2
cosA2
1+
(
2 cos2 A2−1
)
=2 sin
A2
cosA2
2 cos2 A2
=sin
A2
cosA2
= tanA2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
328 Trig or Treat
cosec A tanA2−
cos 2A1+ cos A
≡ 4 sin2 A2
Eyeballing and Mental Gymnastics
1. cosec = 1/sin2. 2A, A/2 suggest expansion using “double angle”, “half angle” formulas3. common denominator4. rearrange and simplify.
LHS = cosec A tanA2−
cos 2A1+ cos A
=1
2 sinA2
cosA2
·
sinA2
cosA2
−
(2 cos2 A−1)
1+ cos A
=1
2 cos2 A2
−
(
2 cos2 A−1)
1+
(
2 cos2 A2−1
)
=1
2 cos2 A2
−
(
2 cos2 A−1)
2 cos2 A2
=1− (2 cos2 A−1)
2 cos2 A2
=2−2 cos2 A
2 cos2 A2
=1− cos2 A
cos2 A2
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 329
=sin2 A
cos2 A2
=
(
2 sinA2
cosA2
)2
cos2 A2
=4 sin2 A
2cos2 A
2
cos2 A2
= 4 sin2 A2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
330 Trig or Treat
sin 2A cos A−2 cos 2A sin A2 sin A− sin 2A
≡ 2 cos2 A2
Eyeballing and Mental Gymnastics
1. 2A suggests “double angle” expansion2. 2 cos2(A/2) on RHS suggest cos A ≡ 2 cos2(A/2)−13. rearrange and simplify.
LHS =sin 2A cos A−2 cos 2A sin A
2 sin A− sin 2A
=2 sin A cos A · cos A−2 cos 2A sin A
2 sin A−2 sin A cos A
=2 sin A(cos2 A− cos 2A)
2 sin A(1− cos A)
=cos2 A− (2 cos2 A−1)
1− cos A
=1− cos2 A1− cos A
=(1− cos A)(1+ cos A)
(1− cos A)
= (1+ cos A)
= 1+
(
2 cos2 A2−1
)
= 2 cos2 A2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 331
cos A1+ cos 2A
+sin A
1− cos 2A≡
sin A+ cos Asin 2A
Eyeballing and Mental Gymnastics
1. 2A suggests expansion of “compound angles”2. rearrange and simplify.
LHS =cos A
1+ cos 2A+
sin A1− cos 2A
=cos A(1− cos 2A)+ sin A(1+ cos 2A)
(1+ cos 2A)(1− cos 2A)
=cos A(1− (2 cos2 A−1))+ sin A(2 cos2 A)
(1− cos2 2A)
=cos A(2−2 cos2 A)+2 sin A cos2 A
sin2 2A
=2 cos A−2 cos3 A+2 sin A cos2 A
sin2 2A
=2 cos A(1− cos2 A+ sin A cos A)
sin2 2A
=2 cos A(sin2 A+ sin A cos A)
sin2 2A
=2 cos A sin A(sin A+ cos A)
sin2 2A
=sin 2A(sin A+ cos A)
sin2 2A
=(sin A+ cos A)
sin 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
332 Trig or Treat
1+ tan A1− tan A
+1− tan A1+ tan A
≡ 2 sec 2A
Eyeballing and Mental Gymnastics
1. t = s/c, sec = 1/cos2. 2A suggests “double angle”3. common denominator4. rearrange and simplify.
LHS =1+ tan A1− tan A
+1− tan A1+ tan A
=(1+ tan A)2 +(1− tan A)2
(1− tan A)(1+ tan A)
=(1+2 tan A+ tan2 A)+(1−2 tan A+ tan2 A)
1− tan2 A
=2+2 tan2 A1− tan2 A
=2(1+ tan2 A)
1− tan2 A
= 2 · (sec2 A)1
1−sin2 Acos2 A
= 2 · sec2 A ·
cos2 A
(cos2 A− sin2 A)
= 21
cos 2A
= 2 sec 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 333
sin 2A+ cos 2A+1sin 2A+ cos 2A−1
≡
tan(45◦ +A)
tan A
Eyeballing and Mental Gymnastics
1. 2A suggests expansion of “compound angle”2. tan(45◦ +A) suggests expansion of “compound angle”3. both sides to be simplified before identity is established4. t = s/c5. rearrange and simplify.
LHS =sin 2A+ cos 2A+1sin 2A+ cos 2A−1
=(2 sin A cos A)+(cos2 A− sin2 A)+(cos2 A+ sin2 A)
(2 sin A cos A)+(cos2 A− sin2 A)− (cos2 A+ sin2 A)
=2 sin A cos A+2 cos2 A
2 sin A cos A−2 sin2 A
=2 cos A(sin A+ cos A)
2 sin A(cos A− sin A)
=1
tan A
(
tan A+11− tan A
)
dividing allterms by cos A.
RHS =tan(45◦ +A)
tan A
=
(
tan 45◦ + tan A1− tan 45◦ tan A
)(
1tan A
)
=1
tan A
(
1+ tan A1− tan A
)
≡ simplified form of LHS
tan 45◦ = 1
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
334 Trig or Treat
√
(
1− sin A1+ sin A
)
≡ sec A− tan A
Eyeballing and Mental Gymnastics
1. square the sq root function2. rearrange and simplify.
LHS =
√
1− sin A1+ sin A
(LHS)2 =1− sin A1+ sin A
=1− sin A1+ sin A
·
(
1− sin A1− sin A
)
=(1− sin A)2
1− sin2 A
=(1− sin A)2
cos2 A
=
(
1cos A
−
sin Acos A
)2
= (sec A− tan A)2
≡ (RHS)2
∴ LHS ≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 335
sin2 2A+2 cos 2A−1
sin2 2A+3 cos 2A−3≡
11− sec 2A
Eyeballing and Mental Gymnastics
1. s2 suggests s2 + c2 ≡ 12. factorisation of LHS3. rearrange and simplify.
LHS =sin2 2A+2 cos 2A−1
sin2 2A+3 cos 2A−3
=(1− cos2 2A)+2 cos 2A−1(1− cos2 2A)+3 cos 2A−3
=cos 2A(2− cos 2A)
3 cos 2A− cos2 2A−2
=cos 2A(2− cos 2A)
(cos 2A−1)(2− cos 2A)
=cos 2A
(cos 2A−1)
=1
1− sec 2A
≡ RHS.
divide numeratorand denominatorby cos 2A
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
336 Trig or Treat
sin(A+45◦)cos(A+45◦)
+cos(A+45◦)sin(A+45◦)
≡ 2 sec 2A
Eyeballing and Mental Gymnastics
1. (A+45◦) suggests expansion of “compound angle”2. 2A suggests “double angle”3. sec = 1/cos4. common denominator5. rearrange and simplify.
LHS =sin(A+45◦)cos(A+45◦)
+cos(A+45◦)sin(A+45◦)
=sin2(A+45◦)+ cos2(A+45◦)
cos(A+45◦)sin(A+45◦)
=1
cos(A+45◦)sin(A+45◦)
=2
2 cos(A+45◦)sin(A+45◦)
=2
sin 2(A+45◦)
=2
sin(2A+90◦)
=2
cos 2A
= 2 sec 2A
≡ RHS.
since sin(x+90)= cos x
(turns out that the problem is easier than expected, as the use of commondenominator resulted in s2 +c2
≡ 1; hence there is no need to expand (A+
45◦)!)
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 337
tan A+ tan 2A ≡
sin A(4 cos2 A−1)
cos A cos 2A
Eyeballing and Mental Gymnastics
1. t = s/c2. 2A suggests expansion of “double angle”3. rearrange and simplify.
LHS = tan A+ tan 2A
=sin Acos A
+sin 2Acos 2A
=sin A cos 2A+ sin 2A cos A
cos A cos 2A
=sin A(2 cos2 A−1)+ cos A(2 sin A cos A)
cos A cos 2A
=sin A(2 cos2 A−1+2 cos2 A)
cos A cos 2A
=sin A(4 cos2 A−1)
cos A cos 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
338 Trig or Treat
(tan A− cosec A)2− (cot A− sec A)2
≡ 2(cosec A− sec A)
Eyeballing and Mental Gymnastics
1. t = s/c, cosec = 1/sin, cot = c/s, sec = 1/cos2. ( )2 suggests s2 + c2 ≡ 13. rearrange and simplify.
LHS = (tan A− cosec A)2− (cot A− sec A)2
= (tan2 A−2 tan Acosec A+ cosec 2A)
− (cot2 A−2 cot A sec A+ sec2 A)
= (tan2 A− sec2 A)+(cosec 2A− cot2 A)
−2sin Acos A
·
1sin A
+2cos Asin A
·
1cos A
= (−1)+(1)−2
cos A+
2sin A
= 2
(
1sin A
−
1cos A
)
= 2(cosec A− sec A)
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 339
2 sin 2A(1−2 sin2 A) ≡ sin 4A
Eyeballing and Mental Gymnastics
1. sin 4A suggests sin 2(2A)2. (1−2 sin2 A) equals cos 2A3. rearrange and simplify.
LHS = 2 sin 2A(1−2 sin2 A)
= 2 sin 2A cos 2A
= sin 4A
≡ RHS.
Alternatively since sin 4A is a standard expression we can proceed with theRHS.
RHS = sin 4A
= 2 sin 2A cos 2A
= 2 sin 2A(1−2 sin2 A)
≡ LHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
340 Trig or Treat
cos 3A ≡ 4 cos3 A−3 cos A
Eyeballing and Mental Gymnastics
1. 3A suggests double expansion of “compound angle”2. rearrange and simplify.
LHS = cos 3A
= cos(2A+A)
= cos 2A cos A− sin 2A sin A
= cos A(cos2 A− sin2 A)− sin A(2 sin A cos A)
= cos3 A− sin2 A cos A−2 sin2 A cos A
= cos3 A−3 sin2 A cos A
= cos3 A−3 cos A(1− cos2 A)
= cos3 A−3 cos A+3 cos3 A
= 4 cos3 A−3 cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 341
32 cos6 A−48 cos4 A+18 cos2 A−1 ≡ cos 6A
Eyeballing and Mental Gymnastics
1. cos 6A on RHS suggest “double angle” formula for (3A)2. 3A suggests expansion of “triple angle”3. this is one of the rare occasions where it may be easier to start with the simpler
RHS4. rearrange and simplify.
RHS = cos 6A
= cos 2(3A)
= 2 cos2(3A)−1
= 2(4 cos3 A−3 cos A)2−1
= 2 cos2 A(4 cos2 A−3)2−1
= 2 cos2 A(16 cos4 A+24 cos2 A+9)−1
= 32 cos6 A+48 cos4 A+18 cos2 A−1
≡ LHS.
see previousproof
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
342 Trig or Treat
cos 4A+4 cos 2A+3 ≡ 8 cos4 A
Eyeballing and Mental Gymnastics
1. cos 4A, cos 2A suggest expansion of “double angle”2. rearrange and simplify.
LHS = (cos 4A)+4 cos 2A+3
= (2 cos2 2A−1)+4 cos 2A+3
= 2 cos2 2A+4 cos 2A+2
= 2(cos2 2A+2 cos 2A+1)
= 2(cos 2A+1)2
= 2(2 cos2 A−1+1)2
= 2(4 cos4 A)
= 8 cos4 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 343
sin 3Asin A
−
cos 3Acos A
≡ 2
Eyeballing and Mental Gymnastics
1. sin 3A, cos 3A suggest expansion of “compound angle”2. rearrange and simplify.
LHS =sin 3Asin A
−
cos 3Acos A
=(3 sin A−4 sin3 A)
sin A−
4(cos3 A−3 cos A)
cos A
=sin A(3−4 sin2 A)
sin A−
cos A(4 cos2 A−3)
cos A
= 3−4 sin2 A−4 cos2 A+3
= 6−4(sin2 A+ cos2 A)
= 6−4
= 2
≡ RHS.
see proofson p. 324and 340
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
344 Trig or Treat
sin4 A+ cos4 A ≡
34
+14
cos 4A
Eyeballing and Mental Gymnastics
1. The cos 4A on the RHS, a standard function, suggests that it may be easier tostart with the RHS through a double expansion of cos 2(2A)
2. rearrange and simplify.
RHS =34
+14
cos 4A
=34
+14
(2 cos2 2A−1)
=34
+12
cos2 2A−
14
=12
+12
(2 cos2 A−1)2
=12
+12
(4 cos4 A−4 cos2 A+1)
= 1+2 cos4 A−2 cos2 A
= cos4 A+ cos4 A−2 cos2 A+1
= cos4 A+(cos2 A−1)2
= cos4 A+(sin2 A)2
= cos4 A+ sin4 A
≡ LHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 345
8 cos4 A−4 cos 2A−3 ≡ cos 4A
Eyeballing and Mental Gymnastics
1. cos 4A on the RHS is a standard expression and suggests the expansion ofcos 4A → cos 2A → cos A
2. hence, easier to begin from the RHS3. rearrange and simplify.
RHS = cos 4A
= cos 2(2A)
= 2 cos2(2A)−1
= 2(2 cos2 A−1)2−1
= 2(4 cos4 A−4 cos2 A+1)−1
= 8 cos4 A−8 cos2 A+2−1
= 8 cos4 A−4(2 cos2 A−1)−2−1
= 8 cos4 A−4 cos 2A−3
≡ LHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
346 Trig or Treat
1−8 sin2 A cos2 A ≡ cos 4A
Eyeballing and Mental Gymnastics
1. cos 4A — standard expression; therefore easier to begin with RHS and expandtwice cos 4A → cos 2A → cos A
2. s2, c2 suggest s2 + c2 ≡ 13. rearrange and simplify.
RHS = cos 4A
= 2 cos2(2A)−1
= 2(2 cos2 A−1)2−1
= 2(4 cos4 A−4 cos2 A+1)−1
= (8 cos4 A−8 cos2 A+2)−1
= 8 cos2 A(cos2 A−1)+1
= 8 cos2 A(−sin2 A)+1
= 1−8 sin2 A cos2 A
≡ LHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy
Level-Two-Games 347
1− cos 2A+ sin Asin 2A+ cos A
≡ tan A
Eyeballing and Mental Gymnastics
1. 2A suggests expansion of “double angle”2. t = s/c3. rearrange and simplify.
LHS =1− cos 2A+ sin A
sin 2A+ cos A
=1− (cos2 A− sin2 A)+ sin A
2 sin A cos A+ cos A
=(cos2 A+ sin2 A)− (cos2 A− sin2 A)+ sin A
cos A(2 sin A+1)
=2 sin2 A+ sin A
cos A(2 sin A+1)
=sin Acos A
(2 sin A+1)
(2 sin A+1)
= tan A
≡ LHS.
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September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-GamesNot-So-Easy Proofs
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
350 Trig or Treat
Angles in a Triangle
(A+B+C) = 180◦
sin A+ sin B+ sin C ≡ 4 cosA2
cosB2
cosC2
Eyeballing and Mental Gymnastics
1. (sin A+ sin B) suggests sin S+ sin T formula2. C = 180◦− (A+B)3. A/2, B/2, C/2 suggest “half angle” formula4. rearrange and simplify.
see next page
September 17, 2007 21:51 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 351
LHS = sin A+sin B+sin C
= 2 sinA+B
2cos
A−B2
+sin C
= 2 sin
(
180◦−C2
)
cos(A−B)
2+2 sin
C2
cosC2
= 2 cosC2
cos(A−B)
2+2 sin
C2
cosC2
= 2 cosC2
(
cosA−B
2+sin
C2
)
= 2 cosC2
(
cosA−B
2+cos
A+B2
)
= 2 cosC2
(
2 cos(A−B)+(A+B)
4cos
(A−B)−(A+B)
4
)
= 2 cosC2
(
2 cos2A4
cos(−2B)
4
)
= 4 cosC2
cosA2
cosB2
= 4 cosA2
cosB2
cosC2
≡RHS.
sin
(
90◦−C2
)
= cosC2
sinC2
= cos
(
90◦−C2
)
= cosA+B
2
|cos(−B) = cos B
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
352 Trig or Treat
sin(A+B+C)
≡
sin A cos B cos C+ sin B cos C cos A+ sin C cos A cos B− sin A sin B sin C
∗
Eyeballing and Mental Gymnastics
1. (A+B+C) suggests expansion of “compound angle” twice2. rearrange and simplify.
LHS = sin(A+B+C)
= sin(A+B)cos C + cos(A+B)sin C
= cos C(sin A cos B+ cos A sin B)
+ sin C(cos A cos B− sin A sin B)
= sin A cos B cos C + sin B cos C cos A
+ sin C cos A cos B− sin A sin B sin C
≡ RHS.
∗Note that:
this general identity is for any three angles A, B and C; they need not necessarilybe for the three angles of a triangle.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 353
Angles in a Triangle
(A+B+C = 180◦)
sin A cos B cos C + sin B cos C cos A+ sin C cos A cos B
≡ sin A sin B sin C
Eyeballing and Mental Gymnastics
1. This identity is easiest proved as a follow-up from the previous proof of thegeneral identity:
sin(A+B+C)≡
sin A cos B cos C
+ sin B cos C cos A
+ sin C cos A cos B
− sin A sin B sin C
2. sin(A+B+C) = sin 180◦ = 0.
Since sin(A+B+C) = sin 180◦ = 0,
∴ sin A cos B cos C
+ sin B cos C cos A
+ sin C cos A cos B
− sin A sin B sin C
= 0
∴ sin A cos B cos C
+ sin B cos C cos A
+ sin C cos A cos B
= sin A sin B sin C
∴ LHS ≡ RHS.
(This identity is extremely valuable for proving many subsequent identities in-volving the three angles of a triangle. Without knowledge of the preceding generalidentity, this identity is more difficult to prove. Try it out for yourself)
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
354 Trig or Treat
Angles in a Triangle
(A+B+C) = 180◦
sin 2A+ sin 2B+ sin 2C ≡ 4 sin A sin B sin C
Eyeballing and Mental Gymnastics
1. sin 2A+ sin 2B suggests sin S+ sin T formula2. sin 2C suggests “double angle” formula3. (A+B+C) = 180◦ therefore express values for (180◦−C) in terms of A+B4. rearrange and simplify.
LHS = sin 2A+ sin 2B+ sin 2C
= 2 sin
(
2A+2B2
)
cos
(
2A−2B2
)
+2 sin C cos C
= 2 sin(A+B)cos(A−B)+2 sin C cos C
= 2 sin C cos(A−B)−2 sin C cos(A+B)
= 2 sin C(cos(A−B)− cos(A+B))
= 2 sin C
(
−2 sin(A−B)+(A+B)
2sin
(A−B)− (A+B)
2
)
= 2 sin C(−2 sin A sin(−B))
= 4 sin C sin A sin B
= 4 sin A sin B sin C
≡ RHS.
see identitiesbelow
sin(A+B) = sin(180◦− (A+B))
= sin C
cos C = −cos(180◦−C)
= −cos(A+B)
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 355
Angles in a Triangle
(A+B+C) = 180◦
sin 2A+ sin 2B+ sin 2C ≡ 4 sin A sin B sin C
For this beautiful identity let’s explore a second approach.
Eyeballing and Mental Gymnastics
1. 2A, 2B, 2C suggest expansion of “double angles”2. A+B+C = 180◦
3. rearrange and simplify.
LHS = sin 2A+ sin 2B+ sin 2C
= 2 sin A cos A
+2 sin B cos B
+2 sin C cos C
= 2 sin A(sin B sin C− cos B cos C)
+2 sin B(sin C sin A− cos C cos A)
+2 sin C(sin A sin B− cos A cos B)
= 6 sin A sin B sin C−2 sin A cos B cos C
−2 sin B cos C cos A
−2 sin C cos A cos B
see identitiesbelow
= 6 sin A sin B sin C−2(sin A sin B sin C)
= 4 sin A sin B sin C
≡ RHS.
from previousproof on p. 353
cos A = cos[180◦− (B+C)]= −cos(B+C)= sin B sin C− cos B cos C
A = 180◦− (B+C)
sin A cos B cos Csin B cos C cos Asin C cos A cos B
=sin(A+B+C)
+ sin A sin B sin Csin(A+B+C)
= sin(180◦)= 0
September 17, 2007 21:51 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
356 Trig or Treat
Angles in a Triangle
(A+B+C) = 180◦
tan A+ tan B+ tan C ≡ tan A tan B tan C
Eyeballing and Mental Gymnastics
1. t = s/c2. (A+B+C) = 180◦
3. rearrange and simplify.
LHS = tan A+ tan B+ tan C
=sin Acos A
+sin Bcos B
+sin Ccos C
=sin A cos B cos C + sin B cos C cos A+ sin C cos A cos B
cos A cos B cos C
=sin A sin B sin C
cos A cos B cos C
= tan A tan B tan C
≡ RHS.
see identity∗
below
∗For (A+B+C) = 180◦
sin A cos B cos C
+sin B cos C cos A
+sin C cos A cos B
= sin A sin B sin Cfrom previousproof on p. 353
An alternative proof is to begin with tan C = − tan(A + B) with expansion oftan(A + B) so that tan A + tan B can be expressed in terms of tan A, tan B andtan C. Try it.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 357
Angles in a Triangle
(A+B+C) = 180◦
cos A+ cos B+ cos C ≡ 4 sinA2
sinB2
sinC2
+1
Eyeballing and Mental Gymnastics
1. cos A+ cos B suggests cos S+ cos T formula2. since (A + B +C) = 180◦, then C/2 is complementary angle of (A + B)/2;
i.e. (90− (A+B)/2)3. rearrange and simplify.
LHS=(cos A+cos B)+(cos C)
=
(
2 cosA+B
2cos
A−B2
)
+
(
1−2 sin2 C2
)
= 2 sinC2
cosA−B
2+1−2 sin2 C
2
= 2 sinC2
(
cosA−B
2−sin
C2
)
+1
= 2 sinC2
(
cosA−B
2−cos
A+B2
)
+1
= 2 sinC2
(
−2 sin12
(
A−B2
+A+B
2
)
sin12
(
A−B2
−A+B2
))
+1
= 2 sinC2
(
−2 sinA2
sin
(−B2
))
+1
= 4 sinC2
sinA2
sinB2
+1
= 4 sinA2
sinB2
sinC2
+1
≡RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
358 Trig or Treat
cos 3A+ cos 2A ≡ 2 cos5A2
cosA2
Eyeballing and Mental Gymnastics
1. cos S+ cos T ≡ 2 cos(S+T)/2 cos(S−T)/22. rearrange and simplify.
LHS = cos 3A+ cos 2A
= 2 cos3A+2A
2cos
3A−2A2
= 2 cos5A2
cosA2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 359
sin 5A− sin 3A ≡ 2 sin A cos 4A
Eyeballing and Mental Gymnastics
1. sin S− sin T = 2 cos(S+T )/2 sin(S−T )/22. rearrange and simplify.
LHS = sin 5A+ sin 3A
= 2 cos5A+3A
2sin
5A−3A2
= 2 cos 4A sin A
= 2 sin A cos 4A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
360 Trig or Treat
sin A+ sin Bsin A− sin B
≡ tanA+B
2cot
A−B2
Eyeballing and Mental Gymnastics
1. sin S+ sin T ≡ 2 sin(S+T )/2 cos(S−T )/22. sin S− sin T ≡ 2 cos(S+T )/2 cos(S−T )/23. rearrange and simplify.
LHS =sin A+ sin Bsin A− sin B
=2 sin
A+B2
cosA−B
2
2 cosA+B
2sin
A−B2
= tanA+B
2cot
A−B2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 361
sin(2A+B)+ sin Bcos(2A+B)+ cos B
≡ tan(A+B)
Eyeballing and Mental Gymnastics
1. sin S+ sin Tcos S+ cos T
2. rearrange and simplify.
LHS =sin(2A+B)+ sin Bcos(2A+B)+ cos B
=2 sin
(2A+B)+B2
cos(2A+B)−B
2
2cos(2A+B)+B
2cos
(2A+B)−B2
=2 sin(A+B)cos A2cos(A+B)cos A
= tan(A+B)
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
362 Trig or Treat
cos A+ cos Bcos A− cos B
≡−cotA+B
2cot
A−B2
Eyeballing and Mental Gymnastics
1. cos S+ cos T2. cos S− cos T3. rearrange and simplify.
LHS =cos A+ cos Bcos A− cos B
=2 cos
A+B2
cosA−B
2
−2 sinA+B
2sin
A−B2
= −cotA+B
2cot
A−B2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 363
sin A+ sin Bcos A+ cos B
≡ tanA+B
2
Eyeballing and Mental Gymnastics
1. sin S+ sin T2. cos S+ cos T3. rearrange and simplify.
LHS =sin A+ sin Bcos A+ cos B
=2 sin
A+B2
cosA−B
2
2 cosA+B
2cos
A−B2
=sin
A+B2
cosA+B
2
= tanA+B
2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
364 Trig or Treat
sin A− sin Bcos A− cos B
≡−cotA+B
2
Eyeballing and Mental Gymnastics
1. sin S− sin T2. cos S− cos T3. rearrange and simplify.
LHS =sin A− sin Bcos A− cos B
=2 cos
A+B2
sinA−B
2
−2 sinA+B
2sin
A−B2
= −cos
A+B2
sinA+B
2
= −cotA+B
2
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 365
sin A+ sin 3A2 sin 2A
≡ cos A
Eyeballing and Mental Gymnastics
1. sin S+ sin T2. rearrange and simplify.
LHS =sin A+ sin 3A
2 sin 2A
=2 sin
A+3A2
cosA−3A
22 sin 2A
=2 sin 2A cos(−A)
2 sin 2A
= cos(−A)
= cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
366 Trig or Treat
sin 4A− sin 2Acos 4A+ cos 2A
≡ tan A
Eyeballing and Mental Gymnastics
1. sin 4A− sin 2A suggests sin S− sin T2. cos 4A+ cos 2A suggests cos S+ cos T3. t = s/c4. rearrange and simplify.
LHS =sin 4A− sin 2Acos 4A+ cos 2A
=2 cos
4A+2A2
sin4A−2A
2
2 cos4A+2A
2cos
4A−2A2
=sin Acos A
= tan A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 367
cos A− cos 3Asin 3A+ sin A
≡ tan A
Eyeballing and Mental Gymnastics
1. cos S− cos T2. sin S+ sin T3. rearrange and simplify.
LHS =cos A− cos 3Asin 3A+ sin A
=−2 sin
A+3A2
sinA−3A
2
2 sinA+3A
2cos
A−3A2
= − sin(−A)
cos(−A)
=sin Acos A
= tan A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
368 Trig or Treat
cos A+ cos 3A2 cos 2A
≡ cos A
Eyeballing and Mental Gymnastics
1. cos S+ cos T2. rearrange and simplify.
LHS =cos A+ cos 3A
2 cos A
=2 cos
A+3A2
cosA−3A
22 cos 2A
=2 cos 2A cos(−A)
2 cos 2A
= cos(−A)
= cos A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 369
sin A− sin 3A
sin2 A− cos2 A≡ 2 sin A
Eyeballing and Mental Gymnastics
1. sin A− sin 3A suggests sin S− sin T formula2. s2
− c2 suggests cos 2A ≡ c2− s2
3. rearrange and simplify.
LHS =sin A− sin 3A
sin2 A− cos2 A
=2 cos
A+3A2
sinA−3A
2−cos 2A
=2 cos 2A sin(−A)
−cos 2A
= 2 sin A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
370 Trig or Treat
sin(A+B)− sin(A−B)
cos(A+B)+ cos(A−B)≡ tan B
Eyeballing and Mental Gymnastics
1. sin S− sin Tcos S− cos T
2. rearrange and simplify.
LHS =sin(A+B)− sin(A−B)
cos(A+B)+ cos(A−B)
=2 cos
(A+B)+(A−B)
2sin
(A+B)− (A−B)
2
2 cos(A+B)+(A−B)
2cos
(A+B)− (A−B)
2
=sin Bcos B
= tan B
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 371
cos A− cos 5Asin 5A+ sin A
≡ tan 2A
Eyeballing and Mental Gymnastics
1. cos S− cos T2. sin S+ sin T3. rearrange and simplify.
LHS =cos A− cos 5Asin 5A+ sin A
=−2 sin
A+5A2
sinA−5A
2
2 sinA+5A
2cos
A−5A2
=−sin(−2A)
cos(−2A)
=sin 2Acos 2A
= tan 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
372 Trig or Treat
sin A+ sin 2A+ sin 3Acos A+ cos 2A+ cos 3A
≡ tan 2A
Eyeballing and Mental Gymnastics
1. sin A + sin 2A + sin 3A, cos A + cos 2A + cos 3A suggest sin S + sin T andcos S+ cos T formulas using A and 3A
2. t = s/c3. rearrange and simplify.
LHS =sin A+ sin 2A+ sin 3A
cos A+ cos 2A+ cos 3A
=sin 2A+(sin A+ sin 3A)
cos 2A+(cos A+ cos 3A)
=
sin 2A+
(
2 sinA+3A
2cos
A−3A2
)
cos 2A+
(
2 cosA+3A
2cos
A−3A2
)
=sin 2A+(2 sin 2A cos(−A))
cos 2A+(2 cos 2A cos(−A))
=3 sin 2A3 cos 2A
=sin 2Acos 2A
= tan 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 373
sin 4A+ sin 2Acos 4A+ cos 2A
≡ tan 3A
Eyeballing and Mental Gymnastics
1. sin S+ sin T2. cos S+ cos T3. rearrange and simplify.
LHS =sin 4A+ sin 2Acos 4A+ cos 2A
=2 sin
4A+2A2
cos4A−2A
2
2 cos4A+2A
2cos
4A−2A2
=sin 3A cos Acos 3A cos A
= tan 3A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
374 Trig or Treat
cos A− cos 3Asin 3A− sin A
≡ tan 2A
Eyeballing and Mental Gymnastics
1. cos S− cos T2. sin S− sin T3. rearrange and simplify.
LHS =cos A− cos 3Asin 3A− sin A
=−2 sin
A+3A2
sinA−3A
2
2 cos3A+A
2sin
3A−A2
= − sin 2A sin(−A)
cos 2A sin A
=sin 2Acos 2A
· sin Asin A
= tan 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 375
sin 4A+ sin 8Acos 4A+ cos 8A
≡ tan 6A
Eyeballing and Mental Gymnastics
1. sin S+ sin T2. cos S+ cos T3. rearrange and simplify.
LHS =sin 4A+ sin 8Acos 4A+ cos 8A
=2 sin
4A+8A2
cos4A−8A
2
2 cos4A+8A
2cos
4A−8A2
=sin 6Acos 6A
= tan 6A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
376 Trig or Treat
sin 4A− sin 8Acos 4A− cos 8A
≡−cot 6A
Eyeballing and Mental Gymnastics
1. sin S− sin T2. cos S+ cos T3. rearrange and simplify.
LHS =sin 4A− sin 8Acos 4A− cos 8A
=2 cos
4A+8A2
sin4A−8A
2
−2 sin4A+8A
2sin
4A−8A2
= −cos 6Asin 6A
= −cot 6A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 377
cos 4A− cos 8Acos 4A+ cos 8A
≡ tan 2A tan 6A
Eyeballing and Mental Gymnastics
1. cos S− cos T2. cos S+ cos T3. rearrange and simplify.
LHS =cos 4A− cos 8Acos 4A+ cos 8A
=−2 sin
4A+8A2
sin4A−8A
2
2 cos4A+8A
2cos
4A−8A2
= − sin 6Acos 6A
· sin(−2A)
cos(−2A)
=sin 6Acos 6A
· sin 2Acos 2A
= tan 2A tan 6A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
378 Trig or Treat
sin 4A+ sin 8Asin 4A− sin 8A
≡ − tan 6Atan 2A
Eyeballing and Mental Gymnastics
1. sin S+ sin T2. sin S− sin T3. rearrange and simplify.
LHS =sin 4A+ sin 8Asin 4A− sin 8A
=2 sin
4A+8A2
cos4A−8A
2
2 cos4A+8A
2sin
4A−8A2
=sin 6Acos 6A
· cos(−2A)
sin(−2A)
= tan 6A ·(
cos 2A−sin 2A
)
=− tan 6A
tan 2A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 379
sin 5A+2 sin 3A+ sin A ≡ 4 sin 3A cos2 A
Eyeballing and Mental Gymnastics
1. sin 5A+2 sin 3A+ sin A suggests sin S+ sin T formula using 5A and A to give3A which is present on both LHS and RHS.
2. rearrange and simplify.
LHS = sin 5A+2 sin 3A+ sin A
= (2 sin 3A)+(sin 5A+ sin A)
= (2 sin 3A)+
(
2 sin5A+A
2cos
5A−A2
)
= 2 sin 3A+2 sin 3A cos 2A
= 2 sin 3A(1+ cos 2A)
= 2 sin 3A(1+(2 cos2 A−1))
= 2 sin 3A(2 cos2 A)
= 4 sin 3A cos2 A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
380 Trig or Treat
1− cos 2A+ cos 4A− cos 6A ≡ 4 sin A cos 2A sin 3A
Eyeballing and Mental Gymnastics
1. cos 4A, cos 6A suggest cos S− cos T2. (1− cos 2A) suggests 1− cos 2A ≡ 2 sin2 A3. rearrange and simplify.
LHS = 1− cos 2A+ cos 4A− cos 6A
= (1− cos 2A)+(cos 4A− cos 6A)
= (2 sin2 A)+
(
−2 sin
(
4A+6A2
)
sin
(
4A−6A2
))
= (2 sin2 A)− (2(sin 5A) sin(−A))
= 2 sin2 A+2 sin A sin 5A
= 2 sin A(sin A+ sin 5A)
= 2 sin A
(
2 sinA+5A
2cos
A−5A2
)
= 4 sin A(sin 3A cos(−2A))
= 4 sin A sin 3A cos 2A
= 4 sin A cos 2A sin 3A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 381
cos 2A− cos 4Acos 2A+ cos 4A
− tan 3A tan A ≡ 0
Eyeballing and Mental Gymnastics
1. cos S− cos T2. cos S+ cos T3. t = s/c4. rearrange and simplify.
LHS =cos 2A− cos 4Acos 2A+ cos 4A
− tan 3A tan A
=
−2 sin
(
2A+4A2
)
sin
(
2A−4A2
)
2 cos
(
2A+4A2
)
cos
(
2A−4A2
) − tan 3A tan A
= − sin 3Acos 3A
· sin(−A)
cos(−A)− tan 3A tan A
= tan 3A tan A− tan 3A tan A
= 0
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
382 Trig or Treat
sin 2A+ sin 4Asin 2A− sin 4A
+tan 3Atan A
≡ 0
Eyeballing and Mental Gymnastics
1. sin S+ sin T2. sin S− sin T3. t = s/c4. rearrange and simplify.
LHS =sin 2A+ sin 4Asin 2A− sin 4A
+tan 3Atan A
=
2 sin
(
2A+4A2
)
cos
(
2A−4A2
)
2 cos
(
2A+4A2
)
sin
(
2A−4A2
) +tan 3Atan A
=sin 3Acos 3A
· cos(−A)
sin(−A)+
tan 3Atan A
= tan 3A ·(
− 1tan A
)
+tan 3Atan A
= − tan 3Atan A
+tan 3Atan A
= 0
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 383
cos 2A− cos 10A ≡ tan 4A(sin 2A+ sin 10A)
Eyeballing and Mental Gymnastics
1. (cos 2A− cos 10A) suggest cos S− cos T2. (sin 2A+ sin 10A) suggests sin S+ sin T3. both sides are complex; ∴ may have to simplify both sides4. tan 4A may come from (10A−2A)/25. rearrange and simplify.
LHS = cos 2A− cos 10A
= −2 sin
(
2A+102
)
sin
(
2A−10A2
)
= −2 sin 6A sin(−4A)
= 2 sin 6A sin 4A.
RHS = tan 4A(sin 2A+ sin 10A)
= tan 4A
(
2 sin
(
2A+102
)
cos
(
2A−10A2
))
= tan 4A(2 sin 6A) cos(−4A))
= tan 4A ·2 sin 6A cos 4A
=sin 4Acos 4A
·2 sin 6A cos 4A
= 2 sin 6A sin 4A
∴ LHS ≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
384 Trig or Treat
sin 2A+ sin 4A− sin 6A ≡ 4 sin A sin 2A sin 3A
Eyeballing and Mental Gymnastics
1. sin 2A− sin 6A suggests sin S− sin T formula2. sin 4A suggests “ double angle” formula to give 2A3. rearrange and simplify.
LHS = sin 2A+(sin 4A)− sin 6A
= 2 cos2A+6A
2sin
2A−6A2
+(sin 4A)
= 2 cos 4A sin(−2A)+ sin 4A
= −2 cos 4A sin 2A+2 sin 2A cos 2A
= 2 sin 2A(cos 2A− cos 4A)
= 2 sin 2A
(
−2 sin2A+4A
2sin
2A−4A2
)
= 2 sin 2A(−2 sin 3A sin(−A))
= 2 sin 2A(2 sin 3A sin A)
= 4 sin A sin 2A sin 3A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 385
cos 3A+2 cos 5A+ cos 7A ≡ 4 cos2 A cos 5A
Eyeballing and Mental Gymnastics
1. cos 3A, cos 7A suggest cos S+ cos T formula2. rearrange and simplify.
LHS = cos 3A+2 cos 5A+ cos 7A
= (cos 3A+ cos 7A)+2 cos 5A
=
(
2 cos3A+7A
2cos
7A−3A2
)
+2 cos 5A
= (2 cos 5A cos 2A)+2 cos 5A
= 2 cos 5A(cos 2A+1)
= 2 cos 5A(2 cos2 A−1+1)
= 4 cos2 A cos 5A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
386 Trig or Treat
1+ cos 2A+ cos 4A+ cos 6A ≡ 4 cos A cos 2A cos 3A
Eyeballing and Mental Gymnastics
1. cos 2A, cos 4A suggest cos S+ cos T formula2. cos 6A suggests cos 2(3A)3. rearrange and simplify.
LHS = 1+ cos 2A+ cos 4A+ cos 6A
= 1+2 cos2A+4A
2cos
2A−4A2
+ cos 6A
= 1+2 cos 3A cos(−A)+ cos 6A
= 1+2 cos 3A cos A+ cos 2(3A)
= 1+2 cos 3A cos A+(2 cos2 3A−1)
= 2 cos 3A cos A+2 cos2 3A
= 2 cos 3A(cos A+ cos 3A)
= 2 cos 3A
(
2 cosA+3A
2· cos
A−3A2
)
= 4 cos 3A cos 2A cos(−A)
= 4 cos A cos 2A cos 3A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 387
1− cos 2A+ cos 4A− cos 6A ≡ 4 sin A cos 2A sin 3A
Eyeballing and Mental Gymnastics
1. cos 2A, cos 4A suggest cos S− cos T2. cos 6A suggests cos 2(3A)3. rearrange and simplify.
LHS = 1− cos 2A+ cos 4A− cos 6A
= 1+(cos 4A− cos 2A)− cos 6A
= 1+
(
−2 sin4A+2A
2sin
4A−2A2
)
− cos 6A
= 1−2 sin 3A sin A− cos 2(3A)
= 1−2 sin 3A sin A− (1−2 sin2 3A)
= −2 sin 3A sin A+2 sin2 3A
= 2 sin 3A(sin 3A− sin A)
= 2 sin 3A
(
2 cos3A+A
2sin
3A−A2
)
= 2 sin 3A(2 cos 2A sin A)
= 4 sin A cos 2A sin 3A
≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
388 Trig or Treat
tan 4A(sin 2A+ sin 10A) ≡ cos 2A− cos 10A
Eyeballing and Mental Gymnastics
1. t = s/c2. sin S+ sin T3. cos S− cos T4. simplify both sides before comparison.
LHS = tan 4A(sin 2A+ sin 10A)
= tan 4A
(
2 sin
(
2A+10A2
)
cos
(
2A−10A2
))
=sin 4Acos 4A
(2 sin 6A cos(−4A))
=sin 4Acos 4A
2 sin 6A cos 4A
= 2 sin 6A sin 4A
RHS = cos 2A− cos 10A
= −2 sin
(
2A+10A2
)
sin
(
2A−10A2
)
= −2 sin 6A sin(−4A)
= 2 sin 6A sin 4A
LHS ≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 389
sin A(sin 3A+ sin 5A) ≡ cos A(cos 3A− cos 5A)
Eyeballing and Mental Gymnastics
1. Both sides are complex; therefore it may be easier to simplify both sides beforecomparison
2. sin S+ sin T3. cos S− cos T4. rearrange and simplify.
LHS = sin A(sin 3A+ sin 5A)
= sin A
(
2 sin3A+5A
2cos
3A−5A2
)
= sin A(2 sin 4A cos(−A))
= 2 sin A cos A sin 4A
RHS = cos A(cos 3A− cos 5A)
= cos A
(
−2 sin3A+5A
2sin
3A−5A2
)
= cos A(−2 sin 4A sin(−A))
= 2 sin A cos A sin 4A
LHS ≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
390 Trig or Treat
sin A(sin A+ sin 3A) ≡ cos A(cos A− cos 3A)
Eyeballing and Mental Gymnastics
1. Both sides appear to be complex and therefore there may be a need to simplifyboth sides before comparison is made
2. sin S+ sin T3. cos S− cos T4. rearrange and simplify.
LHS = sin A(sin A+ sin 3A)
= sin A
(
2 sinA+3A
2cos
A−3A2
)
= sin A(2 sin 2A cos(−A))
= 2 sin A cos A sin 2A
RHS = cos A(cos A− cos 3A)
= cos A
(
−2 sinA+3A
2sin
A−3A2
)
= cos A(−2 sin 2A sin(−A))
= 2 sin A cos A sin 2A
LHS ≡ RHS.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
Level-Three-Games 391
tan A+ tan(A+120◦)+ tan(A+240◦) ≡ 3 tan 3A∗
Eyeballing and Mental Gymnastics
1. tan(A+120◦), tan(A+240◦), tan 3A suggest expansion of “compound angles”2. tan 120◦ = tan(180◦−60◦) =− tan 60◦ = −
√
33. tan 240◦ = tan(180◦+60◦) = tan 60◦ =
√
34. both LHS and RHS are complex; therefore may have to expand both sides for
ease of comparison5. tan 3A suggest tan(2A+A) and tan 2A expansions6. rearrange and simplify.
LHS = tan A+ tan(A+120◦)+ tan(A+240◦)
= tan A+
(
tan A+ tan 120◦
1− tan A tan 120◦
)
+
(
tan A+ tan 240◦
1− tan A tan 240◦
)
= tan A+
(
tan A−√
3
1+√
3 tan A
)
+
(
tan A+√
3
1−√
3 tan A
)
= tan A+(tan A−
√3)(1−
√3 tan A)+(tan A+
√3)(1+
√3 tan A)
(1+√
3 tan A)(1−√
3 tan A)
= tan A+[(tan A−√
3 tan2 A−√
3+3 tan A)
+(tan A+√
3 tan2 A+√
3+3 tan A)]/(1−3 tan2 A)
= tan A+8 tan A
1−3 tan2 A
=tan A(1−3 tan2 A)+8 tan A
1−3 tan2 A
=9 tan A−3 tan3 A
1−3 tan2 A
=3(3 tan A− tan3 A)
1−3 tan2 A
September 17, 2007 21:51 Book:- Trig or Treat (9in x 6in) 11˙not-so-easy
392 Trig or Treat
RHS = 3 tan 3A
= 3 tan(A+2A)
= 3
(
tan A+ tan 2A1− tan A tan 2A
)
= 3
tan A+
(
2 tan A1− tan2 A
)
1− tan A
(
2 tan A1− tan2 A
)
= 3
tan A(1− tan2 A)+2 tan A1− tan2 A
(1− tan2 A)−2 tan2 A1− tan2 A
= 3
(
tan A− tan3 A+2 tan A1− tan2 A−2 tan2 A
)
=3(3 tan A− tan3 A)
1−3 tan2 A
∴ LHS ≡ RHS.
An alternative proof is to write the second and third terms of the LHS in terms ofsin/cos; then add them using common denominator, followed by “compound angleformula” for the numerator, and “cos S+cos T formula” for the denominator. Thissecond proof is shorter and more elegant. Try it out for yourself.∗Note that there is greater beauty if the identity is written as:
tan A+ tan(A+120◦)+ tan(A+240◦) ≡ 3 tan(A+360◦).
Of course, tan(A+360◦) = tan A.
September 17, 2007 21:40 Book:- Trig or Treat (9in x 6in) 12˙addenda
Addenda
September 17, 2007 21:40 Book:- Trig or Treat (9in x 6in) 12˙addenda
394 Trig or Treat
Proof for sin(AA + BB)
O
T1
cos
A
A
sin
A
P
Fig. 1
RO S
B
B
BTQ
1
cos
A
A
sin
A
P
Fig. 2
September 17, 2007 21:40 Book:- Trig or Treat (9in x 6in) 12˙addenda
Addenda 395
Proof for sin(AA + BB)
Let us draw a right-angled triangle with angle A as in Fig. 1 (opposite page)
then : sin A = O/H = sin A/1
cos A = A/H = cos A/1
Let’s add a second triangle with angle B to the first triangle. Figure 2(opposite page)
now : sin(A+B) =OH
=PR1
= PQ+QR (from Fig. 2)From geometry, we know that ]QTO = B (alt ]
′s between // lines)Similarly, ]QPT = B (both are complementary ]s of ]PTQ)
For triangle PQT ,
cos B =PQPT
=PQ
sin A
∴ PQ = sin A cos B
For triangle TOS,
sin B =TSOT
=QR
cos A
∴ QR = cos A sin B
sin(A+B) = PQ+QR
= sin A cos B+ cos A sin B
∴ sin(A+B) ≡ sin A cos B+ cos A sin B
September 17, 2007 21:40 Book:- Trig or Treat (9in x 6in) 12˙addenda
396 Trig or Treat
Proof for cos(AA + BB)
RO S
B
B
BTQ
1
cos
A
A
sin
A
P
September 17, 2007 21:40 Book:- Trig or Treat (9in x 6in) 12˙addenda
Addenda 397
Proof for cos(AA + BB)
From the Fig (opposite page)
cos(A+B) =OROP
=OR1
= OS−RS
]QTO = B (alt ]′s between // lines)
]QPT = ]QTO (both are complementary angles to ]PTQ)
= B
For triangle PQT :
sin B =QTPT
=QT
sin A
∴ QT = sin A sin B
For triangle TOS;
cos B =OSOT
=OS
cos A
∴ OS = cos A cos B
cos(A+B) = OS−RS
= OS−QT
= cos A cos B− sin A sin B
∴ cos(A+B) ≡ cos A cos B− sin A sin B
(since RS=QT)
September 17, 2007 21:40 Book:- Trig or Treat (9in x 6in) 12˙addenda
398 Trig or Treat
Proof for tan(AA + BB)
tan(A+B) =sin(A+B)
cos(A+B)
=sin A cos B+ cos A sin Bcos A cos B− sin A sin B
=tan A+ tan B
1− tan A tan Bdivide allfour terms bycos A cos B
Substitute B by (−B)
tan(A+(−B)) =tan A+ tan(−B)
1− tan A tan(−B)
∴ tan(A−B) =tan A− tan B
1+ tan A tan B
Since tan(−B) = − tan B.
September 17, 2007 21:40 Book:- Trig or Treat (9in x 6in) 12˙addenda
Addenda 399
Appetisers for Higher Trigonometry
sin x = x−x3
3!+
x5
5!−
x7
7!+
x9
9!−·· ·
cos x = 1−x2
2!+
x4
4!−
x6
6!+
x8
8!−·· ·
tan x = x+x3
3+
2x5
15+ · · ·
arcsin x = x−12
x3
3+
1 ·32 ·4
x5
5−
1 ·3 ·52 ·4 ·6
·x7
7+ · · ·
arctan x = x−x3
3+
x5
5−
x7
7+ · · ·
ex = 1+x1!
+x2
2!+
x3
3!+
x4
4!+ · · ·
eix = 1+(ix)1!
+(ix)2
2!+
(ix)3
3!+
(ix)4
4!+ · · ·
=
1−x2
2!+
x4
4!−
x6
6!+
x8
8!−·· ·
+ix1!
−ix3
3!+
ix5
5!−·· ·
eix = cos x+ i sin x
eiπ = cos π + i sin π
= (−1)+ i(0)
eiπ = −1
September 12, 2007 19:19 Book:- Trig or Treat (9in x 6in) 13˙about-author
About the Author
Dr Y E O Adrian graduated from the University of Singapore with firstclass Honours in Chemistry in 1966, and followed up with a Master ofScience degree in 1968.
He received his Master of Arts and his Doctor of Philosophy degreesfrom Cambridge University in 1970, and did post-doctoral research atStanford University, California.
For his research, he was elected Fellow of Christ’s College, Cambridgeand appointed Research Associate at Stanford University in 1970.
His career spans fundamental and applied research and development,academia, and top appointments in politics and industry. His public ser-vice includes philanthropy and sports administration. Among his numer-ous awards are the Charles Darwin Memorial Prize, the Republic ofSingapore’s Distinguished Service Order, the International OlympicCommittee Centenary Medal, and the Honorary Fellowship of Christ’sCollege, Cambridge University.
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