Algebra and Trigonometry

James Stewart, Lothar Redlin, Saleem Watson

Chapter 8

Analytic Trigonometry - all with Video Answers

Educators

Section 1

Trigonometric Identities

Problem 1

An equation is called an identity if it is valid for _____ values of the variable. The equation $2 x=x+x$ is an algebraic identity, and the equation $\sin ^{2} x+\cos ^{2} x=$ _____ is a trigonometric identity.

Audrey Fleming

Audrey Fleming

Numerade Educator

Problem 2

For any $x$ it is true that $\cos (-x)$ has the same value as $\cos x$. We express this fact as the identity _____.

Linh Vu

Numerade Educator

Problem 3

Write the trigonometric expression in terms of sine and cosine, and then simplify.
$$
\cos t \tan t
$$

Linh Vu

Numerade Educator

Problem 4

Write the trigonometric expression in terms of sine and cosine, and then simplify.
$$
\cos t \csc t
$$

Linh Vu

Numerade Educator

Problem 5

Write the trigonometric expression in terms of sine and cosine, and then simplify.
$$
\sin \theta \sec \theta
$$

Linh Vu

Numerade Educator

Problem 6

Write the trigonometric expression in terms of sine and cosine, and then simplify.
$$
\tan \theta \csc \theta
$$

Linh Vu

Numerade Educator

Problem 7

Write the trigonometric expression in terms of sine and cosine, and then simplify.
$$
\tan ^{2} x-\sec ^{2} x
$$

Linh Vu

Numerade Educator

Problem 8

Write the trigonometric expression in terms of sine and cosine, and then simplify.
$$
\frac{\sec x}{\csc x}
$$

Linh Vu

Numerade Educator

Problem 9

Write the trigonometric expression in terms of sine and cosine, and then simplify.
$$
\sin u+\cot u \cos u
$$

Linh Vu

Numerade Educator

Problem 10

Write the trigonometric expression in terms of sine and cosine, and then simplify.
$$
\cos ^{2} \theta\left(1+\tan ^{2} \theta\right)
$$

Linh Vu

Numerade Educator

Problem 11

Write the trigonometric expression in terms of sine and cosine, and then simplify.
$$
\frac{\sec \theta-\cos \theta}{\sin \theta}
$$

Linh Vu

Numerade Educator

Problem 12

Write the trigonometric expression in terms of sine and cosine, and then simplify.
$$
\frac{\cot \theta}{\csc \theta-\sin \theta}
$$

Linh Vu

Numerade Educator

Problem 13

Simplify the trigonometric expression.
$$
\frac{\sin x \sec x}{\tan x}
$$

Linh Vu

Numerade Educator

Problem 14

Simplify the trigonometric expression.
$$
\cos ^{3} x+\sin ^{2} x \cos x
$$

Linh Vu

Numerade Educator

Problem 15

Simplify the trigonometric expression.
$$
\frac{1+\cos y}{1+\sec y}
$$

Linh Vu

Numerade Educator

Problem 16

Simplify the trigonometric expression.
$$
\frac{\tan x}{\sec (-x)}
$$

Linh Vu

Numerade Educator

Problem 17

Simplify the trigonometric expression.
$$
\frac{\sec ^{2} x-1}{\sec ^{2} x}
$$

Linh Vu

Numerade Educator

Problem 18

Simplify the trigonometric expression.
$$
\frac{\sec x-\cos x}{\tan x}
$$

Linh Vu

Numerade Educator

Problem 19

Simplify the trigonometric expression.
$$
\frac{1+\csc x}{\cos x+\cot x}
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 20

Simplify the trigonometric expression.
$$
\frac{\sin x}{\csc x}+\frac{\cos x}{\sec x}
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 21

Simplify the trigonometric expression.
$$
\frac{1+\sin u}{\cos u}+\frac{\cos u}{1+\sin u}
$$

Linh Vu

Numerade Educator

Problem 22

Simplify the trigonometric expression.
$$
\tan x \cos x \csc x
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 23

Simplify the trigonometric expression.
$$
\frac{2+\tan ^{2} x}{\sec ^{2} x}-1
$$

Tanya Chase

Numerade Educator

Problem 24

Simplify the trigonometric expression.
$$
\frac{1+\cot A}{\csc A}
$$

Linh Vu

Numerade Educator

Problem 25

Simplify the trigonometric expression.
$$
\tan \theta+\cos (-\theta)+\tan (-\theta)
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 26

Simplify the trigonometric expression.
$$
\frac{\cos x}{\sec x+\tan x}
$$

Linh Vu

Numerade Educator

Problem 27

Consider the given equation. (a) Verify algebraically that the equation is an identity. (b) Confirm graphically that the equation is an identity.
$$
\frac{\cos x}{\sec x \sin x}=\csc x-\sin x
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 28

Consider the given equation. (a) Verify algebraically that the equation is an identity. (b) Confirm graphically that the equation is an identity.
$$
\frac{\tan y}{\csc y}=\sec y-\cos y
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 29

Verify the identity.
$$
\frac{\sin \theta}{\tan \theta}=\cos \theta
$$

Linh Vu

Numerade Educator

Problem 30

Verify the identity.
$$
\frac{\tan x}{\sec x}=\sin x
$$

Linh Vu

Numerade Educator

Problem 31

Verify the identity.
$$
\frac{\cos u \sec u}{\tan u}=\cot u
$$

Linh Vu

Numerade Educator

Problem 32

Verify the identity.
$$
\frac{\cot x \sec x}{\csc x}=1
$$

CM

Chloe Metzler

Numerade Educator

Problem 33

Verify the identity.
$$
\sin B+\cos B \cot B=\csc B
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 34

Verify the identity.
$$
\cos (-x)-\sin (-x)=\cos x+\sin x
$$

Linh Vu

Numerade Educator

Problem 35

Verify the identity.
$$
\cot (-\alpha) \cos (-\alpha)+\sin (-\alpha)=-\csc \alpha
$$

Linh Vu

Numerade Educator

Problem 36

Verify the identity.
$$
\csc x[\csc x+\sin (-x)]=\cot ^{2} x
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 37

Verify the identity.
$$
\tan \theta+\cot \theta=\sec \theta \csc \theta
$$

AG

Ankit Gupta

Numerade Educator

Problem 38

Verify the identity.
$$
(\sin x+\cos x)^{2}=1+2 \sin x \cos x
$$

Linh Vu

Numerade Educator

Problem 39

Verify the identity.
$$
(1-\cos \beta)(1+\cos \beta)=\frac{1}{\csc ^{2} \beta}
$$

Linh Vu

Numerade Educator

Problem 40

Verify the identity.
$$
\frac{\cos x}{\sec x}+\frac{\sin x}{\csc x}=1
$$

Linh Vu

Numerade Educator

Problem 41

Verify the identity.
$$
\frac{(\sin x+\cos x)^{2}}{\sin ^{2} x-\cos ^{2} x}=\frac{\sin ^{2} x-\cos ^{2} x}{(\sin x-\cos x)^{2}}
$$

Linh Vu

Numerade Educator

Problem 42

Verify the identity.
$$
(\sin x+\cos x)^{4}=(1+2 \sin x \cos x)^{2}
$$

Linh Vu

Numerade Educator

Problem 43

Verify the identity.
$$
\frac{\sec t-\cos t}{\sec t}=\sin ^{2} t
$$

Linh Vu

Numerade Educator

Problem 44

Verify the identity.
$$
\frac{1-\sin x}{1+\sin x}=(\sec x-\tan x)^{2}
$$

Linh Vu

Numerade Educator

Problem 45

Verify the identity.
$$
\frac{1}{1-\sin ^{2} y}=1+\tan ^{2} y
$$

Linh Vu

Numerade Educator

Problem 46

Verify the identity.
$$
\csc x-\sin x=\cos x \cot x
$$

AG

Ankit Gupta

Numerade Educator

Problem 47

Verify the identity.
$$
(\cot x-\csc x)(\cos x+1)=-\sin x
$$

Linh Vu

Numerade Educator

Problem 48

Verify the identity.
$$
\sin ^{4} \theta-\cos ^{4} \theta=\sin ^{2} \theta-\cos ^{2} \theta
$$

Linh Vu

Numerade Educator

Problem 49

Verify the identity.
$$
\left(1-\cos ^{2} X\right)\left(1+\cot ^{2} X\right)=1
$$

Linh Vu

Numerade Educator

Problem 50

Verify the identity.
$$
\cos ^{2} x-\sin ^{2} x=2 \cos ^{2} x-1
$$

AG

Ankit Gupta

Numerade Educator

Problem 51

Verify the identity.
$$
2 \cos ^{2} x-1=1-2 \sin ^{2} x
$$

Linh Vu

Numerade Educator

Problem 52

Verify the identity.
$$
(\tan y+\cot y) \sin y \cos y=1
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 53

Verify the identity.
$$
\frac{1-\cos \alpha}{\sin \alpha}=\frac{\sin \alpha}{1+\cos \alpha}
$$

Linh Vu

Numerade Educator

Problem 54

Verify the identity.
$$
\sin ^{2} \alpha+\cos ^{2} \alpha+\tan ^{2} \alpha=\sec ^{2} \alpha
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 55

Verify the identity.
$$
\tan ^{2} \theta-\sin ^{2} \theta=\tan ^{2} \theta \sin ^{2} \theta
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 56

Verify the identity.
$$
\cot ^{2} \theta \cos ^{2} \theta=\cot ^{2} \theta-\cos ^{2} \theta
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 57

Verify the identity.
$$
\frac{\sin x-1}{\sin x+1}=\frac{-\cos ^{2} x}{(\sin x+1)^{2}}
$$

Linh Vu

Numerade Educator

Problem 58

Verify the identity.
$$
\frac{\sin w}{\sin w+\cos w}=\frac{\tan w}{1+\tan w}
$$

Linh Vu

Numerade Educator

Problem 59

Verify the identity.
$$
\frac{(\sin t+\cos t)^{2}}{\sin t \cos t}=2+\sec t \csc t
$$

Linh Vu

Numerade Educator

Problem 60

Verify the identity.
$$
\sec t \csc t(\tan t+\cot t)=\sec ^{2} t+\csc ^{2} t
$$

Linh Vu

Numerade Educator

Problem 61

Verify the identity.
$$
\frac{1+\tan ^{2} u}{1-\tan ^{2} u}=\frac{1}{\cos ^{2} u-\sin ^{2} u}
$$

Linh Vu

Numerade Educator

Problem 62

Verify the identity.
$$
\frac{1+\sec ^{2} x}{1+\tan ^{2} x}=1+\cos ^{2} x
$$

Linh Vu

Numerade Educator

Problem 63

Verify the identity.
$$
\frac{\sec x}{\sec x-\tan x}=\sec x(\sec x+\tan x)
$$

Linh Vu

Numerade Educator

Problem 64

Verify the identity.
$$
\frac{\sec x+\csc x}{\tan x+\cot x}=\sin x+\cos x
$$

Linh Vu

Numerade Educator

Problem 65

Verify the identity.
$$
\sec v-\tan v=\frac{1}{\sec v+\tan v}
$$

Linh Vu

Numerade Educator

Problem 66

Verify the identity.
$$
\frac{\sin A}{1-\cos A}-\cot A=\csc A
$$

Linh Vu

Numerade Educator

Problem 67

Verify the identity.
$$
\frac{\sin x+\cos x}{\sec x+\csc x}=\sin x \cos x
$$

Linh Vu

Numerade Educator

Problem 68

Verify the identity.
$$
\frac{1-\cos x}{\sin x}+\frac{\sin x}{1-\cos x}=2 \csc x
$$

AG

Ankit Gupta

Numerade Educator

Problem 69

Verify the identity.
$$
\frac{\csc x-\cot x}{\sec x-1}=\cot x
$$

Linh Vu

Numerade Educator

Problem 70

Verify the identity.
$$
\frac{\csc ^{2} x-\cot ^{2} x}{\sec ^{2} x}=\cos ^{2} x
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 71

Verify the identity.
$$
\tan ^{2} u-\sin ^{2} u=\tan ^{2} u \sin ^{2} u
$$

Linh Vu

Numerade Educator

Problem 72

Verify the identity.
$$
\frac{\tan v \sin v}{\tan v+\sin v}=\frac{\tan v-\sin v}{\tan v \sin v}
$$

Linh Vu

Numerade Educator

Problem 73

Verify the identity.
$$
\sec ^{4} x-\tan ^{4} x=\sec ^{2} x+\tan ^{2} x
$$

Linh Vu

Numerade Educator

Problem 74

Verify the identity.
$$
\frac{\cos \theta}{1-\sin \theta}=\sec \theta+\tan \theta
$$

Linh Vu

Numerade Educator

Problem 75

Verify the identity.
$$
\frac{\cos \theta}{1-\sin \theta}=\frac{\sin \theta-\csc \theta}{\cos \theta-\cot \theta}
$$

AG

Ankit Gupta

Numerade Educator

Problem 76

Verify the identity.
$$
\frac{1+\tan x}{1-\tan x}=\frac{\cos x+\sin x}{\cos x-\sin x}
$$

Linh Vu

Numerade Educator

Problem 77

Verify the identity.
$$
\frac{\cos ^{2} t+\tan ^{2} t-1}{\sin ^{2} t}=\tan ^{2} t
$$

Linh Vu

Numerade Educator

Problem 78

Verify the identity.
$$
\frac{1}{1-\sin x}-\frac{1}{1+\sin x}=2 \sec x \tan x
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 79

Verify the identity.
$$
\frac{1}{\sec x+\tan x}+\frac{1}{\sec x-\tan x}=2 \sec x
$$

Linh Vu

Numerade Educator

Problem 80

Verify the identity.
$$
\frac{1+\sin x}{1-\sin x}-\frac{1-\sin x}{1+\sin x}=4 \tan x \sec x
$$

AG

Ankit Gupta

Numerade Educator

Problem 81

Verify the identity.
$$
(\tan x+\cot x)^{2}=\sec ^{2} x+\csc ^{2} x
$$

Linh Vu

Numerade Educator

Problem 82

Verify the identity.
$$
\tan ^{2} x-\cot ^{2} x=\sec ^{2} x-\csc ^{2} x
$$

Linh Vu

Numerade Educator

Problem 83

Verify the identity.
$$
\frac{\sec u-1}{\sec u+1}=\frac{1-\cos u}{1+\cos u}
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 84

Verify the identity.
$$
\frac{\cot x+1}{\cot x-1}=\frac{1+\tan x}{1-\tan x}
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 85

Verify the identity.
$$
\frac{\sin ^{3} x+\cos ^{3} x}{\sin x+\cos x}=1-\sin x \cos x
$$

AG

Ankit Gupta

Numerade Educator

Problem 86

Verify the identity.
$$
\frac{\tan v-\cot v}{\tan ^{2} v-\cot ^{2} v}=\sin v \cos v
$$

Linh Vu

Numerade Educator

Problem 87

Verify the identity.
$$
\frac{1+\sin X}{1-\sin X}=(\tan X+\sec X)^{2}
$$

Linh Vu

Numerade Educator

Problem 88

Verify the identity.
$$
\frac{\tan x+\tan y}{\cot x+\cot y}=\tan x \tan y
$$

Linh Vu

Numerade Educator

Problem 89

Verify the identity.
$$
(\tan x+\cot x)^{4}=\csc ^{4} x \sec ^{4} x
$$

Linh Vu

Numerade Educator

Problem 90

Verify the identity.
$$
(\sin \alpha-\tan \alpha)(\cos \alpha-\cot \alpha)=(\cos \alpha-1)(\sin \alpha-1)
$$

Linh Vu

Numerade Educator

Problem 91

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7$)$ . Assume that $0 \leq \theta<\pi / 2 .$
$$
\frac{x}{\sqrt{1-x^{2}}}, \quad x=\sin \theta
$$

Sheryl Ezze

Numerade Educator

Problem 92

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7$)$ . Assume that $0 \leq \theta<\pi / 2 .$
$$
\sqrt{1+x^{2}}, \quad x=\tan \theta
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 93

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7$)$ . Assume that $0 \leq \theta<\pi / 2 .$
$$
\sqrt{x^{2}-1}, \quad x=\sec \theta
$$

K Joseph

Numerade Educator

Problem 94

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7$)$ . Assume that $0 \leq \theta<\pi / 2 .$
$$
\frac{1}{x^{2} \sqrt{4+x^{2}}}, \quad x=2 \tan \theta
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 95

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7$)$ . Assume that $0 \leq \theta<\pi / 2 .$
$$
\sqrt{9-x^{2}}, \quad x=3 \sin \theta
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 96

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7$)$ . Assume that $0 \leq \theta<\pi / 2 .$
$$
\frac{\sqrt{x^{2}-25}}{x}, \quad x=5 \sec \theta
$$

Norman Atentar

Numerade Educator

Problem 97

Graph $f$ and $g$ in the same viewing rectangle. Do the graphs suggest that the equation $f(x)=g(x)$ is an identity? Prove your answer.
$$
f(x)=\cos ^{2} x-\sin ^{2} x, \quad g(x)=1-2 \sin ^{2} x
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 98

Graph $f$ and $g$ in the same viewing rectangle. Do the graphs suggest that the equation $f(x)=g(x)$ is an identity? Prove your answer.
$$
f(x)=\tan x(1+\sin x), \quad g(x)=\frac{\sin x \cos x}{1+\sin x}
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 99

Graph $f$ and $g$ in the same viewing rectangle. Do the graphs suggest that the equation $f(x)=g(x)$ is an identity? Prove your answer.
$$
f(x)=(\sin x+\cos x)^{2}, \quad g(x)=1
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 100

Graph $f$ and $g$ in the same viewing rectangle. Do the graphs suggest that the equation $f(x)=g(x)$ is an identity? Prove your answer.
$$
f(x)=\cos ^{4} x-\sin ^{4} x, \quad g(x)=2 \cos ^{2} x-1
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 101

Show that the equation is not an identity.
$$
\begin{array}{l}{\text { (a) } \sin 2 x=2 \sin x} & {\text { (b) } \sin (x+y)=\sin x+\sin y} \\ {\text { (c) } \sec ^{2} x+\csc ^{2} x=1} \\ {\text { (d) } \frac{1}{\sin x+\cos x}=\csc x+\sec x}\end{array}
$$

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 102

Cofunction Identities In the right triangle shown, explain why $v=(\pi / 2)-u$ . Explain how you can obtain all six cofunction identities from this triangle for $0 < u < \pi / 2$
Graph cannot copy

Norman Atentar

Numerade Educator

Problem 103

Graphs and Identities Suppose you graph two functions, $f$ and $g,$ on a graphing device and their graphs appear identical in the viewing rectangle. Does this prove that the equation $f(x)=g(x)$ is an identity? Explain.

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 104

Making Up Your Own Identity If you start with a trigonometric expression and rewrite it or simplify it, then setting the original expression equal to the rewritten expression yields a trigonometric identity. For instance, from Example 1 we get the identity
$$
\cos t+\tan t \sin t=\sec t
$$
Use this technique to make up your own identity, then give it to a classmate to verify.

Joshua Eastwood

Joshua Eastwood

Numerade Educator