Introduction In this section, you will study techniques for verifying trigonometric identities.In the next section, you will study techniques for solving trigonometric equa-tions. The key to verifying identities and solving equations is the ability to usethe fundamental identities and the rules of algebra to rewrite trigonometricexpressions.

Remember that a conditional equation is an equation that is true for onlysome of the values in its domain. For example, the conditional equation

Conditional equation

is true only for where is an integer. When you find these values, youare solving the equation.

On the other hand, an equation that is true for all real values in the domainof the variable is an identity. For example, the familiar equation

Identity

is true for all real numbers So, it is an identity.

Verifying Trigonometric IdentitiesAlthough there are similarities, verifying that a trigonometric equation is anidentity is quite different from solving an equation. There is no well-defined setof rules to follow in verifying trigonometric identities, and the process is bestlearned by practice.

Verifying trigonometric identities is a useful process if you need to converta trigonometric expression into a form that is more useful algebraically. Whenyou verify an identity, you cannot assume that the two sides of the equation areequal because you are trying to verify that they are equal. As a result, when verifying identities, you cannot use operations such as adding the same quantityto each side of the equation or cross multiplication.

x.

sin2 x � 1 � cos2 x

nx � n�,

sin x � 0

382 Chapter 5 Analytic Trigonometry

What you should learn• Verify trigonometric identities.

Why you should learn itYou can use trigonometric iden-tities to rewrite trigonometricequations that model real-lifesituations. For instance, inExercise 56 on page 388, you can use trigonometric identitiesto simplify the equation thatmodels the length of a shadowcast by a gnomon (a device usedto tell time).

Verifying Trigonometric Identities

Robert Ginn/PhotoEdit

5.2

You may want to review the distinctionsamong expressions, equations, and identities. Have your students look atsome algebraic identities and conditionalequations before starting this section.It is important for them to understandwhat it means to verify an identity andnot try to solve it as an equation.

Guidelines for Verifying Trigonometric Identities1. Work with one side of the equation at a time. It is often better to work

with the more complicated side first.

2. Look for opportunities to factor an expression, add fractions, square abinomial, or create a monomial denominator.

3. Look for opportunities to use the fundamental identities. Note whichfunctions are in the final expression you want. Sines and cosines pair upwell, as do secants and tangents, and cosecants and cotangents.

4. If the preceding guidelines do not help, try converting all terms to sinesand cosines.

5. Always try something. Even paths that lead to dead ends provide insights.

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Verifying a Trigonometric Identity

Verify the identity

SolutionBecause the left side is more complicated, start with it.

Pythagorean identity

Simplify.

Reciprocal identity

Quotient identity

Simplify.

Notice how the identity is verified. You start with the left side of the equation (themore complicated side) and use the fundamental trigonometric identities tosimplify it until you obtain the right side.

Now try Exercise 5.

There is more than one way to verify an identity. Here is another way toverify the identity in Example 1.

Rewrite as the difference of fractions.

Reciprocal identity

Pythagorean identity

Combining Fractions Before Using Identities

Verify the identity

Solution

Add fractions.

Simplify.

Pythagorean identity

Reciprocal identity

Now try Exercise 19.

� 2 sec2 �

�2

cos2 �

�2

1 � sin2 �

1

1 � sin ��

1

1 � sin ��

1 � sin � � 1 � sin �

�1 � sin ���1 � sin ��

1

1 � sin ��

1

1 � sin �� 2 sec2 �.

� sin2 �

� 1 � cos2 �

sec2 � � 1

sec2 ��

sec2 �

sec2 ��

1

sec2 �

� sin2 �

�sin2 �

�cos2 �� �cos2 ��

� tan2 ��cos2 ��

�tan2 �

sec2 �

sec2 � � 1

sec2 ��

�tan2 � � 1� � 1

sec2 �

sec2 � � 1

sec2 �� sin2 �.

Section 5.2 Verifying Trigonometric Identities 383

Example 1

Example 2

Remember that an identity isonly true for all real values inthe domain of the variable. For instance, in Example 1 the identity is not true when

because is not defined when � � ��2.

sec2 �� � ��2

Encourage your students to identify the reasoning behind each solution step in the examples of this sectionwhile covering the comment lines.This will help students to recognize and remember the fundamentaltrigonometric identities.

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Converting to Sines and Cosines

Verify the identity

SolutionTry converting the left side into sines and cosines.

Quotient identities

Add fractions.

Pythagorean identity

Reciprocal identities

Now try Exercise 29.

Recall from algebra that rationalizing the denominator using conjugates is, onoccasion, a powerful simplification technique. A related form of this technique,shown below, works for simplifying trigonometric expressions as well.

This technique is demonstrated in the next example.

� csc2 x�1 � cos x�

�1 � cos x1 � cos2 x

�1 � cos x

sin2 x1

1 � cos x�

11 � cos x�

1 � cos x1 � cos x�

� sec x csc x �1

cos x �

1

sin x

�1

cos x sin x

�sin2 x � cos2 x

cos x sin x

tan x � cot x �sin x

cos x�

cos x

sin x

tan x � cot x � sec x csc x.

384 Chapter 5 Analytic Trigonometry

Example 4

Verifying Trigonometric Identity

Verify the identity �tan2 x � 1��cos2 x � 1� � �tan2 x.

Example 3

Algebraic SolutionBy applying identities before multiplying, you obtain the following.

Pythagorean identities

Reciprocal identity

Rule of exponents

Quotient identity

Now try Exercise 39.

� �tan2 x

� �� sin x

cos x�2

� �sin2 x

cos2 x

�tan2 x � 1��cos2 x � 1� � �sec2 x���sin2 x�

Numerical SolutionUse the table feature of a graphing utility set inradian mode to create a table that shows the values of and

for different values of as shownin Figure 5.2. From the table you can see that thevalues of and appear to be identical, so

appears tobe an identity.�tan2 x � 1��cos2 x � 1� � �tan2 x

y2y1

x,y2 � �tan2 xy1 � �tan2 x � 1��cos2 x � 1�

Although a graphing utility canbe useful in helping to verify anidentity, you must use algebraictechniques to produce a validproof.

As shown at the right,is considered a

simplified form of because the expression does notcontain any fractions.

1��1 � cos x�csc2 x�1 � cos x�

FIGURE 5.2

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Verifying Trigonometric Identities

Verify the identity

SolutionBegin with the right side, because you can create a monomial denominator bymultiplying the numerator and denominator by

Multiply.

Pythagorean identity

Write as separate fractions.

Simplify.

Identities

Now try Exercise 33.

In Examples 1 through 5, you have been verifying trigonometric identities byworking with one side of the equation and converting to the form given on theother side. On occasion, it is practical to work with each side separately, to obtainone common form equivalent to both sides. This is illustrated in Example 6.

Working with Each Side Separately

Verify the identity

SolutionWorking with the left side, you have

Pythagorean identity

Factor.

Simplify.

Now, simplifying the right side, you have

Write as separate fractions.

Reciprocal identity

The identity is verified because both sides are equal to

Now try Exercise 47.

csc � � 1.

� csc � � 1.

1 � sin �

sin ��

1

sin ��

sin �

sin �

� csc � � 1.

��csc � � 1��csc � � 1�

1 � csc �

cot2 �

1 � csc ��

csc2 � � 1

1 � csc �

cot2 �

1 � csc ��

1 � sin �

sin �.

� sec y � tan y

�1

cos y�

sin y

cos y

�cos y

cos2 y�

cos y sin y

cos2 y

�cos y � cos y sin y

cos2 y

�cos y � cos y sin y

1 � sin2 y

cos y

1 � sin y�

cos y

1 � sin y �1 � sin y

1 � sin y�1 � sin y.

sec y � tan y �cos y

1 � sin y.

Section 5.2 Verifying Trigonometric Identities 385

Example 5

Example 6

Multiply numerator anddenominator by 1 � sin y.

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In Example 7, powers of trigonometric functions are rewritten as morecomplicated sums of products of trigonometric functions. This is a commonprocedure used in calculus.

Three Examples from Calculus

Verify each identity.

a.

b.

c.

Solution

a. Write as separate factors.

Pythagorean identity

Multiply.

b. Write as separate factors.

Pythagorean identity

Multiply.

c. Write as separate factors.

Pythagorean identity

Multiply.

Now try Exercise 49.

� csc2 x�cot x � cot3 x�

� csc2 x�1 � cot2 x� cot x

csc4 x cot x � csc2 x csc2 x cot x

� �cos4 x � cos6 x� sin x

� �1 � cos2 x�cos4 x sin x

sin3 x cos4 x � sin2 x cos4 x sin x

� tan2 x sec2 x � tan2 x

� tan2 x�sec2 x � 1�tan4 x � �tan2 x��tan2 x�

csc4 x cot x � csc2 x�cot x � cot3 x�sin3 x cos4 x � �cos4 x � cos6 x� sin x

tan4 x � tan2 x sec2 x � tan2 x

386 Chapter 5 Analytic Trigonometry

Example 7

Alternative Writing AboutMathematics

a. Ask students to assemble a list oftechniques and strategies forrewriting trigonometric expressionssuch as those demonstrated in theexamples of this section.

b. Ask students to work in pairs. Eachstudent should create an identityequation from the fundamentaltrigonometric identities. Partnersthen trade and verify one another’sidentities. Then have students write abrief explanation of the techniquesthey used to create the identities.

W RITING ABOUT MATHEMATICS

Error Analysis You are tutoring a student in trigonometry. One of the homeworkproblems your student encounters asks whether the following statement is anidentity.

Your student does not attempt to verify the equivalence algebraically, but mistak-enly uses only a graphical approach. Using range settings of

your student graphs both sides of the expression on a graphing utility andconcludes that the statement is an identity.

What is wrong with your student’s reasoning? Explain. Discuss the limitationsof verifying identities graphically.

Yscl � 1Xscl � ��2

Ymax � 20Xmax � 3�

Ymin � �20Xmin � �3�

tan2 x sin2 x �? 5

6 tan2 x

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Section 5.2 Verifying Trigonometric Identities 387

In Exercises 1–38, verify the identity.

1. 2.

3.

4.

5.

6.

7.

8.

9. 10.

11. 12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25. 26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38. sec2��

2� x� � 1 � cot2 x

sin t csc��

2� t� � tan t

sec2 y � cot 2��

2� y� � 1

cos2 � � cos2��

2� �� � 1

�1 � cos �

1 � cos ��

1 � cos �

�sin ��

�1 � sin �

1 � sin ��

1 � sin �

�cos ��

cos x � cos y

sin x � sin y�

sin x � sin y

cos x � cos y� 0

tan x � cot y

tan x cot y� tan y � cot x

tan x � tan y

1 � tan x tan y�

cot x � cot y

cot x cot y � 1

tan x cot xcos x

� sec x

�1 � sin y��1 � sin��y� � cos2 y

csc��x�sec��x�

� �cot x

cos����2� � xsin����2� � x

� tan xtan��

2� �� tan � � 1

cos x �cos x

1 � tan x�

sin x cos x

sin x � cos x

1

sin x � 1�

1

csc x � 1� 1

1 � sin �

cos ��

cos �

1 � sin �� 2 sec �

cos � cot �

1 � sin �� 1 � csc �

1

sin x�

1

csc x� csc x � sin x

1

tan x�

1

cot x� tan x � cot x

sec x � cos x � sin x tan x

csc x � sin x � cos x cot x

sec � � 1

1 � cos �� sec �

1

sec x tan x� csc x � sin x

sec6 x�sec x tan x� � sec4 x�sec x tan x� � sec5 x tan3 x

sin1�2 x cos x � sin5�2 x cos x � cos3 x�sin x

1

tan �� tan � �

sec2 �

tan �

cot2 t

csc t� csc t � sin t

cot3 t

csc t� cos t �csc2 t � 1�

csc2 �

cot �� csc � sec �

cos x � sin x tan x � sec x

sin2 � � sin4 � � cos2 � � cos4 �

cos2 � � sin2 � � 2 cos 2 � � 1

cos2 � � sin2 � � 1 � 2 sin2 �

cot 2 y�sec 2 y � 1� � 1

�1 � sin ���1 � sin �� � cos 2 �

sec y cos y � 1sin t csc t � 1

Exercises 5.2

VOCABULARY CHECK:

In Exercises 1 and 2, fill in the blanks.

1. An equation that is true for all real values in its domain is called an ________.

2. An equation that is true for only some values in its domain is called a ________ ________.

In Exercises 3–8, fill in the blank to complete the trigonometric identity.

3. 4.

5. 6.

7. 8.

PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.

sec��u� � ________csc��u� � ________

cos��

2� u� � ________sin2 u � ________ � 1

cos usin u

� ________1

cot u� ________

333202_0502.qxd 12/5/05 9:01 AM Page 387

In Exercises 39– 46, (a) use a graphing utility to graph eachside of the equation to determine whether the equation isan identity, (b) use the table feature of a graphing utility to determine whether the equation is an identity, and (c) confirm the results of parts (a) and (b) algebraically.

39.

40.

41.

42.

43.

44.

45. 46.

In Exercises 47–50, verify the identity.

47.

48.

49.

50.

In Exercises 51–54, use the cofunction identities to evalu-ate the expression without the aid of a calculator.

51. 52.

53.

54.

55. Rate of Change The rate of change of the functionwith respect to change in the variable

is given by the expression Show thatthe expression for the rate of change can also be

Synthesis

True or False? In Exercises 57 and 58, determine whetherthe statement is true or false. Justify your answer.

57. The equation is an identity,because and

58. The equation is not an identity, because it is true that and

Think About It In Exercises 59 and 60, explain why theequation is not an identity and find one value of thevariable for which the equation is not true.

59.

60.

Skills Review

In Exercises 61–64, perform the operation and simplify.

61. 62.

63. 64.

In Exercises 65–68, use the Quadratic Formula to solve thequadratic equation.

65. 66.

67. 68. 8x2 � 4x � 3 � 03x2 � 6x � 12 � 0

x2 � 5x � 7 � 0x2 � 6x � 12 � 0

�3 � 2i �3��16 �1 � ��4 ��2 � 5i �2�2 � 3i � � ��26

tan � � �sec2 � � 1

sin � � �1 � cos2 �

1 � cot2���6� � 4.1 � tan2���6� � 11

3,1 � tan2 � � 1 � cot2 �

1 � tan2�0� � 1.sin2�0� � cos2�0� � 1sin2 � � cos2 � � 1 � tan2 �

�cos x cot2 x.

cos x � csc x cot x.xf �x� � sin x � csc x

sin2 12 � sin2 40 � sin2 50 � sin2 78

cos2 20 � cos2 52 � cos2 38 � cos2 70

cos2 55 � cos2 35sin2 25 � sin2 65

sin4 x � cos4 x � 1 � 2 cos2 x � 2 cos4 x

cos3 x sin2 x � �sin2 x � sin4 x� cos x

sec4 x tan2 x � �tan2 x � tan4 x� sec2 x

tan5 x � tan3 x sec2 x � tan3 x

cot �

csc � � 1�

csc � � 1

cot �

cos x

1 � sin x�

1 � sin x

cos x

�sin4 � � 2 sin2 � � 1� cos � � cos5 �

csc4 x � 2 csc2 x � 1 � cot4 x

tan 4 x � tan2 x � 3 � sec2 x�4 tan2 x � 3�2 � cos 2 x � 3 cos4 x � sin2 x�3 � 2 cos2 x�

csc x�csc x � sin x��sin x � cos x

sin x� cot x � csc2 x

2 sec2 x � 2 sec2 x sin2 x � sin2 x � cos 2 x � 1

388 Chapter 5 Analytic Trigonometry

56. Shadow Length The length of a shadow cast by avertical gnomon (a device used to tell time) of height when the angle of the sun above the horizon is (seefigure) can be modeled by the equation

h ft

sθ

s �h sin�90 � ��

sin �.

�h

s

Model It

Model It (cont inued)

(a) Verify that the equation for is equal to

(b) Use a graphing utility to complete the table. Letfeet.

(c) Use your table from part (b) to determine the anglesof the sun for which the length of the shadow is thegreatest and the least.

(d) Based on your results from part (c), what time ofday do you think it is when the angle of the sunabove the horizon is 90?

h � 5

h cot �.s

s

5040302010�

s

90807060�

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