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Lecture 33: Section 5.2
Verifying Trigonometric Identities
Verifying trigonometric identities
Conditional equation vs an identity
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Recall: A conditional equation is an equationthat is true for only some of the values in its domain.
ex. tan x = 1
An identity is an equation that is true for all realvalues in its domain.
ex. sin2 x = 1− cos2 x
Guidelines for Verifying Trigonometric
Identities
1. Start with one side. It’s usually easier to startwith the more complicated side.
2. Use known identities. Bring fractional expressionsto a common denominator, factor, and use the fun-damental identities to simplify expressions.
3. Convert to sines and cosines. Sometimes it is help-ful to rewrite all functions in terms of sines and cosines.
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Ex Xt 3 5
TH
JI4
2 9 1 371 330 MATTER WHAT
sin Xtc 2x X VALUE YOUPLUG INTHIS IS TRUE
ex. Verify the trigonometric identities.
1)1
sec x− tan x= secx + tan x
2)sin2 θ
cos θ= sec θ − cos θ
Checkpoint: Lecture 33, problem 1
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tank tl sec2xhits l secxtta f seax tank
secx tanx secxttanx
secxttanxseczxta.mx
secxttan X secxttanx
RHSV12HS
Seco LOSO
L coso Iso Io cosOLOSO
I 050 SinceCOSO Togo LHSV
ex. Verify the trigonometric identity
2 tan x sec x =1
1− sin x−
1
1 + sin x
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RHS I I
1 sin x Itsinx
f i X x l'Iii K nx
1 tsim l sinx
Hsin l sinx Itsinx I sinx
Itsinx I Sinx
l sin'x
2sinx 2sinx054 2tanxsecx LH.SU
ex. Verify the trigonometric identity
cos θ
1− sin θ= sec θ + tan θ
Checkpoint: Lecture 33, problem 2
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HS COSO COSO1 since
Hsiao1 since Itsino
cosolltsinol sin 20
OSO tcososino
050
COTO t.co SinO0 COSIOSO OSO
1cos
t SinoCOSO
Seco tano
RHS
ex. Verify the trigonometric identity
sin θ
1 + cos θ+
1 + cos θ
sin θ= 2 csc θ
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LHS Sino ItcosoItcoso since
SinoItcoso
t H 050 Itcososince Itcoso
Sin O 1 2050 1 CostesinO HCOSO
tSino HCOSO
Sinnott 2 cosotcosIO sin'Otcos 20 1
Sinolltcoso
It 11 2050 2 2050Sinolltcoso Sinolltcoso
z 0 ZnoSino WO
Zosco
RHS V
ex. Verify the trigonometric identity
tan θ + cot θ
sec θ csc θ= 1
Checkpoint: Lecture 33, problem 3
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LHS tano coto singe t Fifasecocscoctosoks.to EEIomm
lssiinfdkinso jtffofoxs.FIcoto sto
Sinksin 20 0520
Sinocogot COS20
Oso Sino cos sinceI 1
cososino cososino
costing cososinI
I LHS V
Sometimes, it is practical to work with each side sepa-rately to obtain one common form equivalent to bothsides.
ex. Verify the trigonometric identity
1 + cos θ
cos θ=
tan2 θ
sec θ − 1
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LHS 11 050 1 t Efg seOSO EOSO
2HS staff seat tantelsecottsecott Seco I
tano secott
isc
Calculus Examples
ex. Verify:
1) sin3 x cos2 x = (cos2 x− cos4 x) sin x
2) sec6 x tan x = (tan x + 2 tan3 x + tan5 x) sec2 x
Checkpoint: Lecture 33, problem 4
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HS sin3Xco5X sinx sin 2 05 1 2 sin 4 1 cosysinx I co5x co5x
Isin x co5x cos4 1RHS V
LHS _sedxtanx
seixlseitxtanx
sedxkseexttanxjyltta.mx xSec2x llttan2x5tanxsec2x I t2tan2xttan4x thanxseax tan x t2tan3x thanx RHS
