In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.

$$

\sin x=\frac{1}{2}, \quad \cos x=\frac{\sqrt{3}}{2}

$$

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In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.

$$

\tan x=\frac{\sqrt{3}}{3}, \quad \cos x=-\frac{\sqrt{3}}{2}

$$

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In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.

$$

\sec \theta=\sqrt{2}, \quad \sin \theta=-\frac{\sqrt{2}}{2}

$$

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In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.

$$

\csc \theta=\frac{25}{7}, \quad \tan \theta=\frac{7}{24}

$$

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In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.

$$

\tan x=\frac{8}{15}, \quad \sec x=-\frac{17}{15}

$$

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In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.

$$

\cot \phi=-3, \quad \sin \phi=\frac{\sqrt{10}}{10}

$$

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In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.

$$

\sec \phi=\frac{3}{2}, \quad \csc \phi=-\frac{3 \sqrt{5}}{5}

$$

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In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.

$$

\cos \left(\frac{\pi}{2}-x\right)=\frac{3}{5}, \quad \cos x=\frac{4}{5}

$$

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In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.

$$

\sin (-x)=-\frac{1}{3}, \quad \tan x=-\frac{\sqrt{2}}{4}

$$

Yara E.

Numerade Educator

In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.

$$

\sec x=4, \quad \sin x>0

$$

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In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.

$$

\tan \theta=2, \quad \sin \theta<0

$$

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In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.

$$

\csc \theta=-5, \quad \cos \theta<0

$$

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In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.

$$

\sin \theta=-1, \quad \cot \theta=0

$$

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In Exercises 11–24, use the given values to evaluate (if possible) all six trigonometric functions.

$$

\tan \theta \text { is undefined, } \sin \theta>0

$$

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In Exercises 25–30, match the trigonometric expression with one of the following.

$\begin{array}{ll}{\text { (a) } \sec x} & {\text { (b) }-1} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}$

$$

\sec x \cos x

$$

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In Exercises 25–30, match the trigonometric expression with one of the following.

$\begin{array}{ll}{\text { (a) } \sec x} & {\text { (b) }-1} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}$

$$

\tan x \csc x

$$

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In Exercises 25–30, match the trigonometric expression with one of the following.

$\begin{array}{ll}{\text { (a) } \sec x} & {\text { (b) }-1} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}$

$$

\cot ^{2} x-\csc ^{2} x

$$

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In Exercises 25–30, match the trigonometric expression with one of the following.

$\begin{array}{ll}{\text { (a) } \sec x} & {\text { (b) }-1} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}$

$$

\left(1-\cos ^{2} x\right)(\csc x)

$$

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In Exercises 25–30, match the trigonometric expression with one of the following.

$\begin{array}{ll}{\text { (a) } \sec x} & {\text { (b) }-1} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}$

$$

\frac{\sin (-x)}{\cos (-x)}

$$

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In Exercises 25–30, match the trigonometric expression with one of the following.

$\begin{array}{ll}{\text { (a) } \sec x} & {\text { (b) }-1} & {\text { (c) } \cot x} \\ {\text { (d) } 1} & {\text { (e) }-\tan x} & {\text { (f) } \sin x}\end{array}$

$$

\frac{\sin [(\pi / 2)-x]}{\cos [(\pi / 2)-x]}

$$

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In Exercises 31–36, match the trigonometric expression with one of the following.

$\begin{array}{ll}{\text { (a) } \csc x} & {\text { (b) } \tan x} & {\text { (c) } \sin ^{2} x} \\ {\text { (d) } \sin x \tan x} & {\text { (e) } \sec ^{2} x} & {\text { (f) } \sec ^{2} x+\tan ^{2} x}\end{array}$

$$

\sin x \sec x

$$

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In Exercises 31–36, match the trigonometric expression with one of the following.

$\begin{array}{ll}{\text { (a) } \csc x} & {\text { (b) } \tan x} & {\text { (c) } \sin ^{2} x} \\ {\text { (d) } \sin x \tan x} & {\text { (e) } \sec ^{2} x} & {\text { (f) } \sec ^{2} x+\tan ^{2} x}\end{array}$

$$

\cos ^{2} x\left(\sec ^{2} x-1\right)

$$

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In Exercises 31–36, match the trigonometric expression with one of the following.

$\begin{array}{ll}{\text { (a) } \csc x} & {\text { (b) } \tan x} & {\text { (c) } \sin ^{2} x} \\ {\text { (d) } \sin x \tan x} & {\text { (e) } \sec ^{2} x} & {\text { (f) } \sec ^{2} x+\tan ^{2} x}\end{array}$

$$

\sec ^{4} x-\tan ^{4} x

$$

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In Exercises 31–36, match the trigonometric expression with one of the following.

$\begin{array}{ll}{\text { (a) } \csc x} & {\text { (b) } \tan x} & {\text { (c) } \sin ^{2} x} \\ {\text { (d) } \sin x \tan x} & {\text { (e) } \sec ^{2} x} & {\text { (f) } \sec ^{2} x+\tan ^{2} x}\end{array}$

$$

\cot x \sec x

$$

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In Exercises 31–36, match the trigonometric expression with one of the following.

$\begin{array}{ll}{\text { (a) } \csc x} & {\text { (b) } \tan x} & {\text { (c) } \sin ^{2} x} \\ {\text { (d) } \sin x \tan x} & {\text { (e) } \sec ^{2} x} & {\text { (f) } \sec ^{2} x+\tan ^{2} x}\end{array}$

$$

\frac{\sec ^{2} x-1}{\sin ^{2} x}

$$

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In Exercises 31–36, match the trigonometric expression with one of the following.

$\begin{array}{ll}{\text { (a) } \csc x} & {\text { (b) } \tan x} & {\text { (c) } \sin ^{2} x} \\ {\text { (d) } \sin x \tan x} & {\text { (e) } \sec ^{2} x} & {\text { (f) } \sec ^{2} x+\tan ^{2} x}\end{array}$

$$

\frac{\cos ^{2}[(\pi / 2)-x]}{\cos x}

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\cot \theta \sec \theta

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\cos \beta \tan \beta

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\tan (-x) \cos x

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\sin x \cot (-x)

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\sin \phi(\csc \phi-\sin \phi)

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\sec ^{2} x\left(1-\sin ^{2} x\right)

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\frac{\cot x}{\csc x}

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\frac{\csc \theta}{\sec \theta}

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\frac{1-\sin ^{2} x}{\csc ^{2} x-1}

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\frac{1}{\tan ^{2} x+1}

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\frac{\tan \theta \cot \theta}{\sec \theta}

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\frac{\sin \theta \csc \theta}{\tan \theta}

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\sec \alpha \cdot \frac{\sin \alpha}{\tan \alpha}

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\frac{\tan ^{2} \theta}{\sec ^{2} \theta}

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\cos \left(\frac{\pi}{2}-x\right) \sec x

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\cot \left(\frac{\pi}{2}-x\right) \cos x

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\frac{\cos ^{2} y}{1-\sin y}

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\cos t\left(1+\tan ^{2} t\right)

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\sin \beta \tan \beta+\cos \beta

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\csc \phi \tan \phi+\sec \phi

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\cot u \sin u+\tan u \cos u

$$

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In Exercises 37–58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

$$

\sin \theta \sec \theta+\cos \theta \csc \theta

$$

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In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\tan ^{2} x-\tan ^{2} x \sin ^{2} x

$$

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In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\sin ^{2} x \csc ^{2} x-\sin ^{2} x

$$

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In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\sin ^{2} x \sec ^{2} x-\sin ^{2} x

$$

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In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\cos ^{2} x+\cos ^{2} x \tan ^{2} x

$$

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In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\frac{\sec ^{2} x-1}{\sec x-1}

$$

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In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\frac{\cos ^{2} x-4}{\cos x-2}

$$

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In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\tan ^{4} x+2 \tan ^{2} x+1

$$

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In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

1-2 \cos ^{2} x+\cos ^{4} x

$$

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In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\sin ^{4} x-\cos ^{4} x

$$

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In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\sec ^{4} x-\tan ^{4} x

$$

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In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\csc ^{3} x-\csc ^{2} x-\csc x+1

$$

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In Exercises 59–70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\sec ^{3} x-\sec ^{2} x-\sec x+1

$$

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In Exercises 71–74, perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

(\sin x+\cos x)^{2}

$$

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In Exercises 71–74, perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

(\cot x+\csc x)(\cot x-\csc x)

$$

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In Exercises 71–74, perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

(2 \csc x+2)(2 \csc x-2)

$$

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In Exercises 71–74, perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

(3-3 \sin x)(3+3 \sin x)

$$

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In Exercises 75–80, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\frac{1}{1+\cos x}+\frac{1}{1-\cos x}

$$

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In Exercises 75–80, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\frac{1}{\sec x+1}-\frac{1}{\sec x-1}

$$

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In Exercises 75–80, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\frac{\cos x}{1+\sin x}+\frac{1+\sin x}{\cos x}

$$

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In Exercises 75–80, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\frac{\tan x}{1+\sec x}+\frac{1+\sec x}{\tan x}

$$

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In Exercises 75–80, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\tan x+\frac{\cos x}{1+\sin x}

$$

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In Exercises 75–80, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$

\tan x-\frac{\sec ^{2} x}{\tan x}

$$

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In Exercises 81–84, rewrite the expression so that it is not in fractional form. There is more than one correct form of each answer.

$$

\frac{\sin ^{2} y}{1-\cos y}

$$

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In Exercises 81–84, rewrite the expression so that it is not in fractional form. There is more than one correct form of each answer.

$$

\frac{5}{\tan x+\sec x}

$$

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In Exercises 81–84, rewrite the expression so that it is not in fractional form. There is more than one correct form of each answer.

$$

\frac{3}{\sec x-\tan x}

$$

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In Exercises 81–84, rewrite the expression so that it is not in fractional form. There is more than one correct form of each answer.

$$

\frac{\tan ^{2} x}{\csc x+1}

$$

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In Exercises $85-88,$ use a graphing utility to complete the table and graph the functions. Make a conjecture about $y_{1}$ and $y_{2}$ .

$$

y_{1}=\cos \left(\frac{\pi}{2}-x\right), \quad y_{2}=\sin x

$$

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In Exercises $85-88,$ use a graphing utility to complete the table and graph the functions. Make a conjecture about $y_{1}$ and $y_{2}$ .

$$

y_{1}=\sec x-\cos x, \quad y_{2}=\sin x \tan x

$$

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In Exercises $85-88,$ use a graphing utility to complete the table and graph the functions. Make a conjecture about $y_{1}$ and $y_{2}$ .

$$

y_{1}=\frac{\cos x}{1-\sin x}, \quad y_{2}=\frac{1+\sin x}{\cos x}

$$

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In Exercises $85-88,$ use a graphing utility to complete the table and graph the functions. Make a conjecture about $y_{1}$ and $y_{2}$ .

$$

y_{1}=\sec ^{4} x-\sec ^{2} x, \quad y_{2}=\tan ^{2} x+\tan ^{4} x

$$

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In Exercises 89–92, use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically.

$$

\cos x \cot x+\sin x

$$

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In Exercises 89–92, use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically.

$$

\sec x \csc x-\tan x

$$

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In Exercises 89–92, use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically.

$$

\frac{1}{\sin x}\left(\frac{1}{\cos x}-\cos x\right)

$$

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In Exercises 89–92, use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically.

$$

\frac{1}{2}\left(\frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}\right)

$$

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In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$

$$

\sqrt{9-x^{2}}, \quad x=3 \cos \theta

$$

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In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$

$$

\sqrt{64-16 x^{2}}, \quad x=2 \cos \theta

$$

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In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$

$$

\sqrt{16-x^{2}}, \quad x=4 \sin \theta

$$

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In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$

$$

\sqrt{49-x^{2}}, \quad x=7 \sin \theta

$$

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In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$

$$

\sqrt{x^{2}-9}, \quad x=3 \sec \theta

$$

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In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$

$$

\sqrt{x^{2}-4}, \quad x=2 \sec \theta

$$

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In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$

$$

\sqrt{x^{2}+25}, \quad x=5 \tan \theta

$$

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In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$

$$

\sqrt{x^{2}+100}, \quad x=10 \tan \theta

$$

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In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$

$$

\sqrt{4 x^{2}+9}, \quad 2 x=3 \tan \theta

$$

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In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$

$$

\sqrt{9 x^{2}+25}, \quad 3 x=5 \tan \theta

$$

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In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$

$$

\sqrt{2-x^{2}}, \quad x=\sqrt{2} \sin \theta

$$

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In Exercises $93-104$ , use trigonometric substitution to write the algebraic expression as a trigonometric function of $\boldsymbol{\theta},$ where $0<\boldsymbol{\theta}<\pi / 2$

$$

\sqrt{10-x^{2}}, \quad x=\sqrt{10} \sin \theta

$$

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In Exercises $105-108,$ use the trigonometric substitution to write the algebraic equation as a trigonometric equation of $\theta$ where $-\pi / 2<\theta<\pi / 2 .$ Then find $\sin \theta$ and $\cos \theta$.

$$

3=\sqrt{9-x^{2}}, \quad x=3 \sin \theta

$$

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In Exercises $105-108,$ use the trigonometric substitution to write the algebraic equation as a trigonometric equation of $\theta$ where $-\pi / 2<\theta<\pi / 2 .$ Then find $\sin \theta$ and $\cos \theta$.

$$

3=\sqrt{36-x^{2}}, \quad x=6 \sin \theta

$$

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In Exercises $105-108,$ use the trigonometric substitution to write the algebraic equation as a trigonometric equation of $\theta$ where $-\pi / 2<\theta<\pi / 2 .$ Then find $\sin \theta$ and $\cos \theta$.

$$

2 \sqrt{2}=\sqrt{16-4 x^{2}}, \quad x=2 \cos \theta

$$

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In Exercises $105-108,$ use the trigonometric substitution to write the algebraic equation as a trigonometric equation of $\theta$ where $-\pi / 2<\theta<\pi / 2 .$ Then find $\sin \theta$ and $\cos \theta$.

$$

-5 \sqrt{3}=\sqrt{100-x^{2}}, \quad x=10 \cos \theta

$$

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In Exercises $109-112,$ use a graphing utility to solve the equation for $\theta,$ where $0 \leq \theta<2 \pi$

$$

\sin \theta=\sqrt{1-\cos ^{2} \theta}

$$

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In Exercises $109-112,$ use a graphing utility to solve the equation for $\theta,$ where $0 \leq \theta<2 \pi$

$$

\cos \theta=-\sqrt{1-\sin ^{2} \theta}

$$

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In Exercises $109-112,$ use a graphing utility to solve the equation for $\theta,$ where $0 \leq \theta<2 \pi$

$$

\sec \theta=\sqrt{1+\tan ^{2} \theta}

$$

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In Exercises $109-112,$ use a graphing utility to solve the equation for $\theta,$ where $0 \leq \theta<2 \pi$

$$

\csc \theta=\sqrt{1+\cot ^{2} \theta}

$$

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In Exercises 113–118, rewrite the expression as a single logarithm and simplify the result.

$$

\ln |\cos x|-\ln |\sin x|

$$

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In Exercises 113–118, rewrite the expression as a single logarithm and simplify the result.

$$

\ln |\sec x|+\ln |\sin x|

$$

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In Exercises 113–118, rewrite the expression as a single logarithm and simplify the result.

$$

\ln |\sin x|+\ln |\cot x|

$$

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In Exercises 113–118, rewrite the expression as a single logarithm and simplify the result.

$$

\ln |\tan x|+\ln |\csc x|

$$

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In Exercises 113–118, rewrite the expression as a single logarithm and simplify the result.

$$

\ln |\cot t|+\ln \left(1+\tan ^{2} t\right)

$$

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In Exercises 113–118, rewrite the expression as a single logarithm and simplify the result.

$$

\ln \left(\cos ^{2} t\right)+\ln \left(1+\tan ^{2} t\right)

$$

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In Exercises $119-122,$ use a calculator to demonstrate the identity for each value of $\theta$.

$\csc ^{2} \theta-\cot ^{2} \theta=1$

$\begin{array}{ll}{\text { (a) } \theta=132^{\circ}} & {\text { (b) } \theta=\frac{2 \pi}{7}}\end{array}$

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In Exercises $119-122,$ use a calculator to demonstrate the identity for each value of $\theta$.

$\tan ^{2} \theta+1=\sec ^{2} \theta$

(a) $\theta=346^{\circ} \quad$ (b) $\theta=3.1$

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In Exercises $119-122,$ use a calculator to demonstrate the identity for each value of $\theta$.

$\cos \left(\frac{\pi}{2}-\theta\right)=\sin \theta$

$\begin{array}{ll}{\text { (a) } \theta=80^{\circ}} & {\text { (b) } \theta=0.8}\end{array}$

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In Exercises $119-122,$ use a calculator to demonstrate the identity for each value of $\theta$.

$\sin (-\theta)=-\sin \theta$

$\begin{array}{lll}{\text { (a) } \theta=250^{\circ}} & {\text { (b) } \theta=\frac{1}{2}}\end{array}$

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The forces acting on an object weighing $W$ units on an inclined plane positioned at an angle of

$\theta$ with the horizontal (see figure) are modeled by

$$\mu W \cos \theta=W \sin \theta$$

where $\mu$ is the coefficient of friction. Solve the equation for $\mu$ and simplify the result.

Ishita J.

Numerade Educator

The rate of change of the function $f(x)=-x+\tan x$ is given by the expression $-1+\sec ^{2} x .$ Show that that expression can also be written as tan' $x$ .

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The rate of change of the function $f(x)=\sec x+\cos x$ is given by the expression $\sec x \tan x-\sin x .$ Show that this expression can also be written as $\sin x \tan ^{2} x .$

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The rate of change of the function $f(x)=-\csc x-\sin x$ is given by the expression $\csc x \cot x-\cos x .$ Show that this expression can also be written as $\cos x \cot ^{2} x$

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In Exercises 127 and 128, determine whether the statement is true or false. Justify your answer.

The even and odd trigonometric identities are helpful for determining whether the value of a trigonometric function is positive or negative.

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In Exercises 127 and 128, determine whether the statement is true or false. Justify your answer.

A cofunction identity can be used to transform a tangent function so that it can be represented by a

cosecant function.

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In Exercises $129-132,$ fill in the blanks. (Note: The notation $x \rightarrow c^{+}$ indicates that $x$ approaches $c$ from the right and $x \rightarrow c^{-}$ indicates that $x$ approaches $c$ from the left.

As $x \rightarrow \frac{\pi^{-}}{2}, \sin x \rightarrow \qquad$ and $\csc x \rightarrow$

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In Exercises $129-132,$ fill in the blanks. (Note: The notation $x \rightarrow c^{+}$ indicates that $x$ approaches $c$ from the right and $x \rightarrow c^{-}$ indicates that $x$ approaches $c$ from the left.

As $x \rightarrow 0^{+}, \cos x \rightarrow \qquad$ and and $\sec x \rightarrow$

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In Exercises $129-132,$ fill in the blanks. (Note: The notation $x \rightarrow c^{+}$ indicates that $x$ approaches $c$ from the right and $x \rightarrow c^{-}$ indicates that $x$ approaches $c$ from the left.

As $x \rightarrow \frac{\pi}{2}, \tan x \rightarrow \qquad$ and $\cot x \rightarrow$

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In Exercises $129-132,$ fill in the blanks. (Note: The notation $x \rightarrow c^{+}$ indicates that $x$ approaches $c$ from the right and $x \rightarrow c^{-}$ indicates that $x$ approaches $c$ from the left.

As $x \rightarrow \pi^{+}, \sin x \rightarrow \qquad$ and $\csc x \rightarrow$

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In Exercises 133–138, determine whether or not the equation is an identity, and give a reason for your answer.

$$

\cos \theta=\sqrt{1-\sin ^{2} \theta}

$$

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In Exercises 133–138, determine whether or not the equation is an identity, and give a reason for your answer.

$$

\cot \theta=\sqrt{\csc ^{2} \theta+1}

$$

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In Exercises 133–138, determine whether or not the equation is an identity, and give a reason for your answer.

$\frac{(\sin k \theta)}{(\cos k \theta)}=\tan \theta, \quad k$ is a constant.

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In Exercises 133–138, determine whether or not the equation is an identity, and give a reason for your answer.

$$

\frac{1}{(5 \cos \theta)}=5 \sec \theta

$$

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In Exercises 133–138, determine whether or not the equation is an identity, and give a reason for your answer.

$$

\sin \theta \csc \theta=1

$$

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In Exercises 133–138, determine whether or not the equation is an identity, and give a reason for your answer.

$$

\csc ^{2} \theta=1

$$

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Use the trigonometric substitution $u=a \sin \theta,$ where $-\pi / 2<\theta<\pi / 2$ and $a>0,$ to simplify the expression $\sqrt{a^{2}-u^{2}}$

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Use the trigonometric substitution $u=a \tan \theta,$ where $-\pi / 2<\theta<\pi / 2$ and $a>0,$ to simplify the expression $\sqrt{a^{2}+u^{2}}$

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Use the trigonometric substitution $u=a \sec \theta,$ where $0<\theta<\pi / 2$ and $a>0,$ to simplify the expression $\sqrt{u^{2}-a^{2}}$

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(a) Use the definitions of sine and cosine to derive the Pythagorean identity $\sin ^{2} \theta+\cos ^{2} \theta=1$

(b) Use the Pythagorean identity $\sin ^{2} \theta+\cos ^{2} \theta=1$ to derive the other Pythagorean identities, $1+\tan ^{2} \theta=\sec ^{2} \theta$ and $1+\cot ^{2} \theta=\csc ^{2} \theta$ Discuss how to remember these identities and other fundamental identities.

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