464 CHAPTER 7 Analytic Trigonometry

In this section we establish some additional identities involving trigonometricfunctions. But first, we review the definition of an identity.

DEFINITION Two functions and are said to be identically equal if

for every value of x for which both functions are defined. Such an equation isreferred to as an identity. An equation that is not an identity is called aconditional equation.

f1x2 = g1x2gf

114. Bending Light The speed of yellow sodium light(wavelength, 589 nanometers) in a certain liquid ismeasured to be meters per second. What is theindex of refraction of this liquid, with respect to air, forsodium light?*[Hint: The speed of light in air is approximately

meters per second.]

115. Bending Light A beam of light with a wavelength of 589 nanometers traveling in air makes an angle of incidenceof 40° on a slab of transparent material, and the refractedbeam makes an angle of refraction of 26°. Find the index ofrefraction of the material.*

116. Bending Light A light ray with a wavelength of 589 nanometers (produced by a sodium lamp) travelingthrough air makes an angle of incidence of 30° on a smooth,flat slab of crown glass. Find the angle of refraction.*

2.998 * 108

1.92 * 108

117. A light beam passes through a thick slab of material whoseindex of refraction is Show that the emerging beam isparallel to the incident beam.*

118. Brewster’s Law If the angle of incidence and the angle ofrefraction are complementary angles, the angle of incidenceis referred to as the Brewster angle . The Brewster angleis related to the index of refractions of the two media,and , by the equation , where is the index of refraction of the incident medium and is theindex of refraction of the refractive medium. Determine theBrewster angle for a light beam traveling through water (at20°C) that makes an angle of incidence with a smooth, flatslab of crown glass.

n2

n1n1 sin uB = n2 cos uBn2

n1

uB

n2 .

119. Explain in your own words how you would use your calcula-tor to solve the equation Howwould you modify your approach to solve the equationcot x = 5, 0 6 x 6 2p?

cos x = - 0.6, 0 … x 6 2p.120. Provide a justification as to why no further points of

intersection (and therefore solutions) exist in Figure 25 onpage 459 for or x 7 4p.x 6 - p

Explaining Concepts: Discussion and Writing

1. 2. 3. 4. 5. 6. {0.76}e 0, 52fe 1 - 25

2,

1 + 252

fe - 1, 54f22

2; - 1

2e 3

2f

‘Are You Prepared?’ Answers

* Adapted from Halliday and Resnick, Fundamentals of Physics, 7th ed.,2005, John Wiley & Sons.

Now Work the ‘Are You Prepared?’ problems on page 469.

OBJECTIVES 1 Use Algebra to Simplify Trigonometric Expressions (p. 465)

2 Establish Identities (p. 466)

7.4 Trigonometric Identities

• Fundamental Identities (Section 6.3, p. 385) • Even–Odd Properties (Section 6.3, p. 389)

PREPARING FOR THIS SECTION Before getting started, review the following:

Quotient Identities

tan u = sin ucos u cot u = cos u

sin u

Reciprocal Identities

csc u = 1sin u sec u = 1

cos u cot u = 1

tan u

Pythagorean Identities

cot2 u + 1 = csc2 u

sin2 u + cos2 u = 1 tan2 u + 1 = sec2 u

Even–Odd Identities

csc1 - u2 = - csc u sec1 - u2 = sec u cot1 - u2 = - cot u

sin1 - u2 = - sin u cos1 - u2 = cos u tan1 - u2 = - tan u

SECTION 7.4 Trigonometric Identities 465

For example, the following are identities:

The following are conditional equations:

True only if

True only if an integer

True only if or an integer

Below is a list of the trigonometric identities that we have established thusfar.

x = 5p4

+ 2kp, kx = p4

+ 2kp sin x = cos x

x = kp, k sin x = 0

x = - 52

2x + 5 = 0

1x + 122 = x2 + 2x + 1 sin2 x + cos2

x = 1 csc x = 1sin x

This list of identities comprises what we shall refer to as the basic trigonometricidentities. These identities should not merely be memorized, but should be known(just as you know your name rather than have it memorized). In fact, minor variationsof a basic identity are often used. For example, we might want to use

instead of For this reason, among others, you need to knowthese relationships and be comfortable with variations of them.

sin2 u + cos2 u = 1.

sin2 u = 1 - cos2 u or cos2 u = 1 - sin2 u

Use Algebra to Simplify Trigonometric ExpressionsThe ability to use algebra to manipulate trigonometric expressions is a key skill thatone must have to establish identities. Some of the techniques that are used inestablishing identities are multiplying by a “well-chosen 1,” writing a trigonometricexpression over a common denominator, rewriting a trigonometric expression interms of sine and cosine only, and factoring.

1

466 CHAPTER 7 Analytic Trigonometry

(a)

(b)

.

(c)

(d)

Now Work P R O B L E M S 9 , 1 1 , A N D 1 3

sin2 v - 1tan v sin v - tan v

=1sin v + 121sin v - 12

tan v1sin v - 12 = sin v + 1tan v

= cos u + cos usin u cos u

= 2 cos usin u cos u

= 2sin u

æ cot u = cos usin u

= cos u + sin u cos u + cot u sin u - cos u sin usin u cos u

=cos u + cos u

sin u# sin u

sin u cos u

1 + sin usin u

+ cot u - cos ucos u

= 1 + sin usin u

# cos ucos u

+ cot u - cos ucos u

# sin usin u

=cos u11 - sin u2

cos2 u= 1 - sin u

cos u

æ Multiply by a well-chosen 1: 1 - sin u1 - sin u

cos u

1 + sin u= cos u

1 + sin u# 1 - sin u1 - sin u

=cos u11 - sin u2

1 - sin2 u

cot ucsc u

=

cos usin u

1sin u

= cos usin u

# sin u1

= cos uSolution

Establish IdentitiesIn the examples that follow, the directions will read “Establish the identity. . . . ” Asyou will see, this is accomplished by starting with one side of the given equation(usually the one containing the more complicated expression) and, using appropriatebasic identities and algebraic manipulations, arriving at the other side. The selectionof appropriate basic identities to obtain the desired result is learned only throughexperience and lots of practice.

2

Using Algebraic Techniques to Simplify Trigonometric Expressions

(a) Simplify by rewriting each trigonometric function in terms of sine and

cosine functions.

(b) Show that by multiplying the numerator and denominator

by

(c) Simplify by rewriting the expression over a common

denominator.

(d) Simplify by factoring.sin2 v - 1

tan v sin v - tan v

1 + sin usin u

+ cot u - cos ucos u

1 - sin u.

cos u1 + sin u

= 1 - sin ucos u

cot ucsc u

EXAMPLE 1

Establishing an Identity

Establish the identity: csc u# tan u = sec u

EXAMPLE 2

!

NOTE A graphing utility can be used toprovide evidence of an identity. Forexample, if we graph and the graphs appear tobe the same. This provides evidencethat However, it does notprove their equality. A graphing utilitycannot be used to establish anidentity—identities must be establishedalgebraically. "

Y1 = Y2 .

Y2 = sec u,Y1 = csc u# tan u

Start with the left side, because it contains the more complicated expression, andapply a reciprocal identity and a quotient identity.

Having arrived at the right side, the identity is established.

Now Work P R O B L E M 1 9

csc u# tan u = 1 sin u

# sin u

cos u= 1

cos u= sec u

Solution

Begin with the left side and, because the arguments are apply Even–OddIdentities.

Even–Odd Identities

Pythagorean Identity = 1

= 1sin u22 + 1cos u22 = 1 - sin u22 + 1cos u22 sin21 - u2 + cos21 - u2 = 3sin1 - u242 + 3cos1 - u242- u,Solution

We begin with two observations: The left side contains the more complicatedexpression. Also, the left side contains expressions with the argument whereasthe right side contains expressions with the argument We decide, therefore, tostart with the left side and apply Even–Odd Identities.

Even–Odd Identities

Simplify.

Factor.

Cancel and simplify. = cos u - sin u

=1sin u - cos u21 sin u + cos u 2

- 1 sin u + cos u 2 =1sin u22 - 1cos u22

- sin u - cos u

=1 - sin u22 - 1cos u22

- sin u - cos u

sin21 - u2 - cos21 - u2sin1 - u2 - cos1 - u2 =

3sin1 - u242 - 3cos1 - u242sin1 - u2 - cos1 - u2

u.- u,

Solution

=tan u1 1 + tan u 2

tan u + 1 = tan u

1 + tan u1 + cot u

= 1 + tan u

1 + 1tan u

= 1 + tan utan u + 1

tan u

Solution

Now Work P R O B L E M S 2 3 A N D 2 7

SECTION 7.4 Trigonometric Identities 467

Establishing an Identity

Establish the identity: sin21 - u2 + cos21 - u2 = 1

EXAMPLE 3

Establishing an Identity

Establish the identity:sin21 - u2 - cos21 - u2sin1 - u2 - cos1 - u2 = cos u - sin u

EXAMPLE 4

Establishing an Identity

Establish the identity:1 + tan u1 + cot u

= tan u

EXAMPLE 5

!

!

!

!

468 CHAPTER 7 Analytic Trigonometry

When sums or differences of quotients appear, it is usually best to rewrite themas a single quotient, especially if the other side of the identity consists of only oneterm.

The left side is more complicated, so we start with it and proceed to add.

Add the quotients.

= sin2 u + 1 + 2 cos u + cos2 u11 + cos u21sin u2

sin u1 + cos u

+ 1 + cos usin u

=sin2 u + 11 + cos u2211 + cos u21sin u2

Solution

Remove parentheses inthe numerator.

Regroup.

Pythagorean Identity

Factor and cancel.

Reciprocal Identity

Now Work P R O B L E M 4 9

Sometimes it helps to write one side in terms of sine and cosine functions only.

= 2 csc u

= 2sin u

=21 1 + cos u 21 1 + cos u 21sin u2

= 2 + 2 cos u11 + cos u21sin u2 =1sin2 u + cos2 u2 + 1 + 2 cos u11 + cos u21sin u2

tan v + cot v

sec v csc v=

sin vcos v

+ cos vsin v

1cos v

# 1sin v

=

sin2 v + cos2 vcos v sin v

1cos v sin v

Solution

ææChange to sinesand cosines.

Add the quotients in the numerator.

Divide the quotients;sin2 v + cos2 v = 1.

æ

= 1cos v sin v

# cos v sin v1

= 1

Now Work P R O B L E M 6 9

Sometimes, multiplying the numerator and denominator by an appropriate factor will result in a simplification.

Establishing an Identity

Establish the identity:sin u

1 + cos u+ 1 + cos u

sin u= 2 csc u

EXAMPLE 6

Establishing an Identity

Establish the identity:tan v + cot v

sec v csc v= 1

EXAMPLE 7

!

!

Start with the left side and multiply the numerator and the denominator by (Alternatively, we could multiply the numerator and denominator of the right sideby )

1 - sin u

cos u= 1 - sin u

cos u# 1 + sin u1 + sin u

1 - sin u.

1 + sin u.Solution

Multiply the numerator anddenominator by 1 + sin u.

Cancel. = cos u1 + sin u

1 - sin2 u = cos2 u = cos2 ucos u11 + sin u2

= 1 - sin2 ucos u11 + sin u2

Now Work P R O B L E M 5 3

Although a lot of practice is the only real way to learn how to establish identities,the following guidelines should prove helpful.

Guidelines for Establishing Identities1. It is almost always preferable to start with the side containing the more

complicated expression.2. Rewrite sums or differences of quotients as a single quotient.3. Sometimes rewriting one side in terms of sine and cosine functions only

will help.4. Always keep your goal in mind.As you manipulate one side of the expression,

you must keep in mind the form of the expression on the other side.

‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

1. True or False (p. 385)sin2 u = 1 - cos2 u. 2. True or False .(p. 389)

sin 1 - u2 + cos 1 - u2 = cos u - sin u

7.4 Assess Your Understanding

WARNING Be careful not to handleidentities to be established as if theywere conditional equations. You cannotestablish an identity by such methodsas adding the same expression to eachside and obtaining a true statement.This practice is not allowed, becausethe original statement is precisely theone that you are trying to establish.You do not know until it has beenestablished that it is, in fact, true. "

3. Suppose that and are two functions with the samedomain. If for every x in the domain, theequation is called a(n) . Otherwise, it is called a(n)

equation.

4.5. cos1 - u2 - cos u = .

tan2 u - sec2 u = .

f1x2 = g1x2gf 6. True or False for any value of

7. True or False In establishing an identity, it is often easiestto just multiply both sides by a well-chosen nonzero expressioninvolving the variable.

8. True or False for any u Z 12k + 12 p

2.tan u# cos u = sin u

u.sin1 - u2 + sin u = 0

Concepts and Vocabulary

9. Rewrite in terms of sine and cosine functions:

10. Rewrite in terms of sine and cosine functions:

cot u# sec u.

tan u# csc u.11. Multiply by

12. Multiply by 1 - cos u1 - cos u

.sin u

1 + cos u

1 + sin u1 + sin u

.cos u

1 - sin u

Skill BuildingIn Problems 9–18, simplify each trigonometric expression by following the indicated direction.

SECTION 7.4 Trigonometric Identities 469

Establishing an Identity

Establish the identity:1 - sin u

cos u= cos u

1 + sin u

EXAMPLE 8

!

470 CHAPTER 7 Analytic Trigonometry

In Problems 19–98, establish each identity.

31. cos2 u11 + tan2 u2 = 1 32. 11 - cos2 u211 + cot2 u2 = 1 33. 1sin u + cos u22 + 1sin u - cos u22 = 2

19. csc u# cos u = cot u 20. sec u# sin u = tan u 21. 1 + tan21 - u2 = sec2 u

22. 1 + cot21 - u2 = csc2 u 23. cos u1tan u + cot u2 = csc u 24. sin u1cot u + tan u2 = sec u

25. tan u cot u - cos2 u = sin2 u 26. sin u csc u - cos2 u = sin2 u 27. 1sec u - 121sec u + 12 = tan2 u

28. 1csc u - 121csc u + 12 = cot2 u 29. 1sec u + tan u21sec u - tan u2 = 1 30. 1csc u + cot u21csc u - cot u2 = 1

46.csc u - 1

cot u= cot u

csc u + 147.

1 + sin u1 - sin u

= csc u + 1csc u - 1

48.cos u + 1cos u - 1

= 1 + sec u1 - sec u

49.1 - sin v

cos v+ cos v

1 - sin v= 2 sec v 50.

cos v1 + sin v

+ 1 + sin vcos v

= 2 sec v 51.sin u

sin u - cos u= 1

1 - cot u

52. 1 - sin2 u1 + cos u

= cos u 53.1 - sin u1 + sin u

= 1sec u - tan u22 54.1 - cos u1 + cos u

= 1csc u - cot u22

57. tan u + cos u1 + sin u

= sec u 58.sin u cos u

cos2 u - sin2 u= tan u

1 - tan2 u59.

tan u + sec u - 1tan u - sec u + 1

= tan u + sec u

60.sin u - cos u + 1sin u + cos u - 1

= sin u + 1cos u

61.tan u - cot utan u + cot u

= sin2 u - cos2 u 62.sec u - cos usec u + cos u

= sin2 u1 + cos2 u

63.tan u - cot utan u + cot u

+ 1 = 2 sin2 u 64.tan u - cot utan u + cot u

+ 2 cos2 u = 1 65.sec u + tan ucot u + cos u

= tan u sec u

66.sec u

1 + sec u= 1 - cos u

sin2 u 67.1 - tan2 u1 + tan2 u

+ 1 = 2 cos2 u 68.1 - cot2 u1 + cot2 u

+ 2 cos2 u = 1

69.sec u - csc u

sec u csc u= sin u - cos u 70.

sin2 u - tan ucos2 u - cot u

= tan2 u 71. sec u - cos u = sin u tan u

72. tan u + cot u = sec u csc u 73.1

1 - sin u+ 1

1 + sin u= 2 sec2 u 74.

1 + sin u1 - sin u

- 1 - sin u1 + sin u

= 4 tan u sec u

75.sec u

1 - sin u= 1 + sin u

cos3 u76.

1 + sin u1 - sin u

= 1sec u + tan u22 77.1sec v - tan v22 + 1

csc v1sec v - tan v2 = 2 tan v

55. 56.cot u

1 - tan u+ tan u

1 - cot u= 1 + tan u + cot u

cos u1 - tan u

+ sin u1 - cot u

= sin u + cos u

34. tan2 u cos2 u + cot2 u sin2 u = 1 35. sec4 u - sec2 u = tan4 u + tan2 u 36. csc4 u - csc2 u = cot4 u + cot2 u

37. sec u - tan u = cos u1 + sin u

38. csc u - cot u = sin u1 + cos u

39. 3 sin2 u + 4 cos2 u = 3 + cos2 u

40. 9 sec2 u - 5 tan2 u = 5 + 4 sec2 u 41. 1 - cos2 u1 + sin u

= sin u 42. 1 - sin2 u1 - cos u

= - cos u

43.1 + tan v1 - tan v

= cot v + 1cot v - 1

44.csc v - 1csc v + 1

= 1 - sin v1 + sin v

45.sec ucsc u

+ sin ucos u

= 2 tan u

13. Rewrite over a common denominator:

14. Rewrite over a common denominator:

15. Multiply and simplify:1sin u + cos u21sin u + cos u2 - 1

sin u cos u

11 - cos v

+ 11 + cos v

sin u + cos ucos u

+ cos u - sin usin u

16. Multiply and simplify:

17. Factor and simplify:

18. Factor and simplify:cos2 u - 1

cos2 u - cos u

3 sin2 u + 4 sin u + 1sin2 u + 2 sin u + 1

1tan u + 121tan u + 12 - sec2 u

tan u

78. 79.sin u + cos u

cos u- sin u - cos u

sin u= sec u csc usec2 v - tan2 v + tan v

sec v= sin v + cos v

80. 81.sin3 u + cos3 usin u + cos u

= 1 - sin u cos usin u + cos u

sin u- cos u - sin u

cos u= sec u csc u

82. 83.cos2 u - sin2 u

1 - tan2 u= cos2 u

sin3 u + cos3 u1 - 2 cos2 u

= sec u - sin utan u - 1

84. 85.12 cos2 u - 122cos4 u - sin4 u

= 1 - 2 sin2 ucos u + sin u - sin3 u

sin u= cot u + cos2 u

86. 87.1 + sin u + cos u1 + sin u - cos u

= 1 + cos usin u

1 - 2 cos2 usin u cos u

= tan u - cot u

88. 89. 1a sin u + b cos u22 + 1a cos u - b sin u22 = a2 + b21 + cos u + sin u1 + cos u - sin u

= sec u + tan u

90. 91.tan a + tan bcot a + cot b

= tan a tan b12a sin u cos u22 + a21cos2 u - sin2 u22 = a2

92. 1tan a + tan b211 - cot a cot b2 + 1cot a + cot b211 - tan a tan b2 = 0

93. 1sin a + cos b22 + 1cos b + sin a21cos b - sin a2 = 2 cos b1sin a + cos b294. 1sin a - cos b22 + 1cos b + sin a21cos b - sin a2 = - 2 cos b1sin a - cos b295. 96. ln ƒtan uƒ = ln ƒsin uƒ - ln ƒcos uƒln ƒsec uƒ = - ln ƒcos uƒ

97. 98. ln ƒsec u + tan uƒ + ln ƒsec u - tan uƒ = 0ln ƒ1 + cos uƒ + ln ƒ1 - cos uƒ = 2 ln ƒsin uƒ

In Problems 99–102, show that the functions f and g are identically equal.

99. 100. f1x2 = cos x # cot x g1x2 = csc x - sin xf1x2 = sin x # tan x g1x2 = sec x - cos x

101. 102. f1u2 = tan u + sec u g1u2 = cos u1 - sin u

f1u2 = 1 - sin ucos u

- cos u1 + sin u g1u2 = 0

Applications and Extensions

103. Searchlights A searchlight at the grand opening of a newcar dealership casts a spot of light on a wall located 75 metersfrom the searchlight.The acceleration of the spot of light isfound to be . Show that this is

equivalent to .

Source: Adapted from Hibbeler, Engineering Mechanics:Dynamics, 10th ed. © 2004

r.. = 1200 a1 + sin2 ucos3 u

br.. = 1200 sec u12 sec2 u - 12r..

104. Optical Measurement Optical methods of measurementoften rely on the interference of two light waves. If twolight waves, identical except for a phase lag, are mixedtogether, the resulting intensity, or irradiance, is given by

Show that this is

equivalent to .

Source: Experimental Techniques, July/August 2002

It = 12A cos u22It = 4A2(csc u - 1)(sec u + tan u)csc u sec u

.

Explaining Concepts: Discussion and Writing

105. Write a few paragraphs outlining your strategy forestablishing identities.

106. Write down the three Pythagorean Identities.

107. Why do you think it is usually preferable to start with theside containing the more complicated expression whenestablishing an identity?

108. Make up an identity that is not a Fundamental Identity.

‘Are You Prepared?’ Answers

1. True 2. True

SECTION 7.4 Trigonometric Identities 471