Deck 6: Analytic Trigonometry

Question 1

Complete the identity.
$\frac { \csc x \cot x } { \sec x } = ?$

A) $\cot ^ { 2 } x$
B) 1
C) $\csc ^ { 2 } x$
D) $\sec ^ { 2 } x$

Answer 1

A

Question 2

Complete the identity.
$$\frac { \sin x } { \cos x } + \frac { \cos x } { \sin x } = ?$$

A) $\sec x \csc x$
B) $- 2 \tan ^ { 2 } x$
C) $1 + \cot x$
D) $\sin x \tan x$

Answer 2

A

Question 3

Complete the identity.
$\frac { ( \csc x + 1 ) ( \csc x - 1 ) } { \cot ^ { 2 } x } = ?$

A) 1
B) 2
C) $- 1$
D) 0 )0

Answer 3

A

Question 4

Complete the identity.
$\frac { \cos ^ { 2 } x - \sin ^ { 2 } x } { 1 - \tan ^ { 2 } x } = ?$

A) $\cos ^ { 2 } x$
B) $\sin ^ { 2 } x$
C) 1
D) $- 1$

Question 5

Complete the identity.
$\cot ^ { 2 } 2 x + \cos ^ { 2 } 2 x + \sin ^ { 2 } 2 x =$ ?

A) $\csc ^ { 2 } 2 x$
B) $\sin ^ { 2 } 2 x$
C) $\cos ^ { 2 } 2 x$
D) 2

Question 6

Complete the identity.
$\frac { \cos x - \sin x } { \cos x } + \frac { \sin x - \cos x } { \sin x } = ?$

A) $2 - \sec x \csc x$
B) $1 - \sec x \csc x$
C) $2 + \sec x \csc x$
D) $\sec x \csc x$

Question 7

Complete the identity.
$$\frac { 1 - \sin x } { \cos x } = ?$$

A) $\sec x - \tan x$
B) $\sec x + \tan x$
C) $- \sec x - \tan x$
D) $\sec x - \tan x + 1$

Question 8

Complete the identity.
$2 \tan x - ( 1 + \tan x ) ^ { 2 } = ?$

A) $- \sec ^ { 2 } x$
B) 1
C) 0
D) $1 - \sin x$

Question 9

Complete the identity.
$\tan x \cdot \cot x =$ ?

A) 1
B) $- 1$
C) 0
D) $\sin x$

Question 10

Complete the identity.
$$1 - \frac { \sin ^ { 2 } x } { 1 + \cos x } = \text { ? }$$

A) $\cos x$
B) 0
C) $\tan x$
D) $\cot x$

Question 11

Complete the identity.
$\tan x ( \cot x - \cos x ) =$ ?

A) $1 - \sin x$
B) $- \sec ^ { 2 } x$
C) 1
D) 0

Question 12

Complete the identity.
$\frac { \sin x + \cos x } { \sin x } - \frac { \cos x - \sin x } { \cos x } = ?$

A) $\sec x \csc x$
B) $1 - \sec x \csc x$
C) $2 + \sec x \csc x$
D) $2 - \sec x \csc x$

Question 13

Complete the identity.
$\csc x ( \sin x + \cos x ) =$ ?

A) $1 + \cot x$
B) $\sin x \tan x$
C) $- 2 \tan ^ { 2 } x$
D) $\sec x \csc x$

Question 14

Complete the identity.
$\sin ^ { 2 } x + \sin ^ { 2 } x \cot ^ { 2 } x =$ ?

A) 1
B) $\sin ^ { 2 } x + 1$
C) $\cot ^ { 2 } x + 1$
D) $\cot ^ { 2 } x - 1$

Question 15

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

A) 1
B) 0
$C ) - \sec ^ { 2 } x$
D) $1 - \sin x$

Question 16

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

A) 2
B) 1
C) 0
D) $\tan x$

Question 17

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

A) $3 \sec ^ { 4 } x - 2$
B) $4 \sec ^ { 4 } x$
C) $\sec ^ { 4 } x + 2$
D) $\tan ^ { 2 } x - 1$

Question 18

Complete the identity.
$\sec x - \frac { 1 } { \sec x } = ?$

A) $\sin x \tan x$
B) $1 + \cot x$
C) $- 2 \tan ^ { 2 } x$
D) $\sec x \csc x$

Question 19

Complete the identity.
$\sin ^ { 4 } x - \cos ^ { 4 } x =$ ?

A) $1 - 2 \cos ^ { 2 } x$
B) $1 + 2 \cos ^ { 2 } x$
C) $1 + 2 \sin ^ { 2 } x$
D) $1 - 2 \sin ^ { 2 } x$

Question 20

Complete the identity.
$\sin ^ { 2 } x + \tan ^ { 2 } x + \cos ^ { 2 } x = ?$

A) $\sec ^ { 2 } x$
B) $\tan ^ { 2 } x$
C) $\cot ^ { 3 } x$
D) $\sin x$

Question 21

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

Question 22

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

Question 23

Verify the identity.
$$\left( 1 + \tan ^ { 2 } u \right) \left( 1 - \sin ^ { 2 } u \right) = 1$$

Question 24

Verify the identity.
$$\cot ^ { 2 } x + \csc ^ { 2 } x = 2 \csc ^ { 2 } x - 1$$

Question 25

Use the graph to complete the identity.
$\cos ^ { 2 } x + \cos x - 1 + \sin ^ { 2 } x ; \cos x$

A) $\cos x$
B) $- \cos x$
C) $2 + \cos x$
D) $2 \cos x$

Question 26

Verify the identity.
$$\tan \theta \cdot \csc \theta = \sec \theta$$

Question 27

Use the graph to complete the identity.
$\csc x + \tan ^ { 2 } x \csc x ; \cos x$ and $\sin x$

A)
$\frac { 1 } { \sin x \cos ^ { 2 } x }$
B) $\frac { \sin x + \cos x } { \sin x \cos x }$
C) $\frac { 1 } { \sin x \cos x }$
D) $\cos x - \sin x$

Question 28

Complete the identity.
$\csc ^ { 2 } x \sec x =$ ?

A) $\sec x + \csc x \cot x$
B) $\sec x - \csc x \cot x$
C) $\csc x \cot x - \sec x$
D) $\sec x + \csc x$

Question 29

Verify the identity.
$$1 + \sec ^ { 2 } x \sin ^ { 2 } x = \sec ^ { 2 } x$$

Question 30

Use the graph to complete the identity.
$$\frac { \sec ^ { 2 } x \csc x } { \sec ^ { 2 } x + \csc ^ { 2 } x } = ?$$

A) $\sin x$
B) $\cos x$
C) $\csc x$
D) $\sec x$

Question 31

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal.
$\cos x - \cos x \sin x = \cos ^ { 3 } x$

A) $\frac { \pi } { 4 }$
B) $\frac { \pi } { 2 }$
C) $\pi$
D) 0

Question 32

Use the Formula for the Cosine of the Difference of Two Angles
$\cos \left( 255 ^ { \circ } - 15 ^ { \circ } \right)$

A) $- \frac { 1 } { 2 }$
B) $- \frac { \sqrt { 3 } } { 2 }$
C) $\frac { 17 } { 4 }$
D) $\frac { \sqrt { 3 } } { 2 }$

Question 33

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal.
$\sin ( x + \pi ) = \sin x$

A) $\frac { \pi } { 2 }$
B) 0
C) $2 \pi$
D) $\pi$

Question 34

Use the Formula for the Cosine of the Difference of Two Angles
$\cos \left( \frac { 5 \pi } { 18 } - \frac { \pi } { 9 } \right)$

A) $\frac { \sqrt { 3 } } { 2 }$
B) $\frac { 1 } { 2 }$
C) $\frac { 1 } { 4 }$
D) 1 Identify α and β in the following expression which is the right side of the formula for cos (α - β).

Question 35

Use the graph to complete the identity.
$\frac { ( \sec x + \tan x ) ( \sec x - \tan x ) } { \sec x } =$ ?

A) $\cos x$
B) $\sin x$
C) $\csc x$
D) $\sec x$

Question 36

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal.
$\cos ( x + \pi ) = \cos x$

A) 0
B) $\frac { \pi } { 2 }$
C) $- \frac { \pi } { 2 }$
D) $\frac { 3 \pi } { 2 }$

Question 37

Verify the identity.
$$\cot \theta \cdot \sec \theta = \csc \theta$$

Question 38

Use the graph to complete the identity.
$\frac { \cos x \tan x - 4 \tan x + 5 \cos x - 20 } { \tan x + 5 } =$ ?

A) $\cos x - 4$
B) $\cos x + 4$
C) $\sin x - 5$
D) $\sin x + 5 \cos x$

Question 39

Use the graph to complete the identity.
$$\frac { 1 + \cos x } { \sin x } + \frac { \sin x } { 1 + \cos x } = ?$$

A) $2 \csc x$
B) $2 \sec x$
C) $2 \sin x$
D) $2 \cos x$ Rewrite the expression in terms of the given function or functions.

Question 40

Use the graph to complete the identity.
$( \sec x + \csc x ) ( \sin x + \cos x ) - 2 - \cot x ; \tan x$

A) $\tan x$
B) $2 + \tan x$
C) $2 \tan x$
D) 0

Question 41

Use Sum and Difference Formulas for Cosines and Sines
$\sin 255 ^ { \circ }$

A) $- \frac { \sqrt { 2 } ( \sqrt { 3 } + 1 ) } { 4 }$
B) $- \frac { \sqrt { 2 } ( \sqrt { 3 } - 1 ) } { 4 }$
C) $\frac { \sqrt { 2 } ( \sqrt { 3 } + 1 ) } { 4 }$
D) $\frac { \sqrt { 2 } ( \sqrt { 3 } - 1 ) } { 4 }$

Question 42

Use the given information to find the exact value of the expression.
$\sin \alpha = \frac { 24 } { 25 } , \alpha$ lies in quadrant II, and $\cos \beta = \frac { 2 } { 5 } , \beta$ lies in quadrant $\mathrm { I } \quad$ Find $\cos ( \alpha - \beta )$ .

A) $\frac { - 14 + 24 \sqrt { 21 } } { 125 }$
B) $\frac { 48 + 7 \sqrt { 21 } } { 125 }$
C) $\frac { 48 - 7 \sqrt { 21 } } { 125 }$
D) $\frac { 14 - 24 \sqrt { 21 } } { 125 }$

Question 43

Use Sum and Difference Formulas for Cosines and Sines
$\sin 25 ^ { \circ } \cos 35 ^ { \circ } + \cos 25 ^ { \circ } \sin 35 ^ { \circ }$

A) $\frac { \sqrt { 3 } } { 2 }$
B) $\frac { 1 } { 2 }$
C) $\frac { 5 } { 12 }$
D) $\frac { \sqrt { 3 } } { 3 }$

Question 44

Use Sum and Difference Formulas for Cosines and Sines
$\cos \frac { 7 \pi } { 12 } \sin \frac { 5 \pi } { 12 } - \cos \frac { 5 \pi } { 12 } \sin \frac { 7 \pi } { 12 }$

A) $\frac { 1 } { 2 }$
B) $\frac { \sqrt { 3 } } { 2 }$
C) $\frac { 1 } { 4 }$
D) 1

Question 45

Complete the identity.
$\cos \left( x + \frac { \pi } { 2 } \right) =$ ?

A) $- \sin x$
B) $\sin x$
C) $\cos x$
D) $- \cos x$

Question 46

Use the Formula for the Cosine of the Difference of Two Angles
$\cos \left( 160 ^ { \circ } \right) \cos \left( 40 ^ { \circ } \right) + \sin \left( 160 ^ { \circ } \right) \sin \left( 40 ^ { \circ } \right)$

A) $\cos \left( 120 ^ { \circ } \right)$
B) $\cos \left( 210 ^ { \circ } \right)$
C) $\cos \left( 190 ^ { \circ } \right)$
D) $\cos \left( 220 ^ { \circ } \right)$

Question 47

Complete the identity.
$\cos ( \alpha + \beta ) + \cos ( \alpha - \beta ) =$ ?

A) $2 \cos \alpha \cos \beta$
B) $\cos \alpha \cos \beta$
C) $2 \sin \alpha \cos \beta$
D) $\sin \beta \cos \alpha$

Question 48

Complete the identity.
$\frac { \cos ( \alpha - \beta ) } { \cos \alpha \sin \beta } =$ ?

A) $\tan \alpha + \cot \beta$
B) $\tan \alpha + \cot \alpha$
C) $\tan \alpha + \tan \beta$
D) $\cot \beta + \tan \alpha \tan \beta$

Question 49

Use the Formula for the Cosine of the Difference of Two Angles
$\cos \left( \frac { 7 \pi } { 18 } \right) \cos \left( \frac { 2 \pi } { 9 } \right) + \sin \left( \frac { 7 \pi } { 18 } \right) \sin \left( \frac { 2 \pi } { 9 } \right)$

A) $\alpha = \frac { 7 \pi } { 18 } , \beta = \frac { 2 \pi } { 9 }$
B) $\alpha = \frac { 7 \pi } { 18 } , \beta = - \frac { 2 \pi } { 9 }$
C) $\alpha = - \frac { 2 \pi } { 9 } , \beta = \frac { 7 \pi } { 18 }$
D) $\alpha = \frac { 2 \pi } { 9 } , \beta = \frac { 7 \pi } { 18 }$ Write the expression as the cosine of an angle, knowing that the expression is the right side of the formula for $$\cos ( \alpha - \beta ) \text { with particular values for } \alpha \text { and } \beta .$$

Question 50

Use the Formula for the Cosine of the Difference of Two Angles
$\cos \left( 155 ^ { \circ } \right) \cos \left( 35 ^ { \circ } \right) + \sin \left( 155 ^ { \circ } \right) \sin \left( 35 ^ { \circ } \right)$

A) $- \frac { 1 } { 2 }$
B) $- \frac { \sqrt { 3 } } { 2 }$
C) $- \sqrt { 3 }$
D) $- 2$

Question 51

Use Sum and Difference Formulas for Cosines and Sines
$\sin 195 ^ { \circ } \cos 75 ^ { \circ } - \cos 195 ^ { \circ } \sin 75 ^ { \circ }$

A) $\frac { \sqrt { 3 } } { 2 }$
B) $- \frac { 1 } { 2 }$
C) $\frac { 13 } { 4 }$
D) $- \frac { \sqrt { 3 } } { 2 }$

Question 52

Use Sum and Difference Formulas for Cosines and Sines
$\cos \left( 30 ^ { \circ } + 45 ^ { \circ } \right)$

A) $\frac { \sqrt { 2 } ( \sqrt { 3 } - 1 ) } { 4 }$
B) $\frac { \sqrt { 2 } ( \sqrt { 3 } + 1 ) } { 4 }$
C) $- \frac { \sqrt { 2 } ( \sqrt { 3 } - 1 ) } { 4 }$
D) $- \frac { \sqrt { 2 } ( \sqrt { 3 } + 1 ) } { 4 }$

Question 53

Complete the identity.
$\cos \left( x - \frac { 11 \pi } { 6 } \right) =$ ?

A) $\frac { 1 } { 2 } ( \sqrt { 3 } \cos x - \sin x )$
B) $\frac { \sqrt { 3 } } { 2 } ( \cos x - \sin x )$
C) $- \frac { \sqrt { 3 } } { 2 } ( \cos x + \sin x )$
D) $- \frac { \sqrt { 3 } } { 2 } ( \cos x - \sin x )$

Question 54

Use Sum and Difference Formulas for Cosines and Sines
$\cos 285 ^ { \circ }$

A) $\frac { \sqrt { 2 } ( \sqrt { 3 } - 1 ) } { 4 }$
B) $- \frac { \sqrt { 2 } ( \sqrt { 3 } - 1 ) } { 4 }$
C) $- \sqrt { 2 } ( \sqrt { 3 } + 1 )$
D) $- \sqrt { 2 } ( \sqrt { 3 } - 1 )$ Find the exact value of the expression.

Question 55

Use Sum and Difference Formulas for Cosines and Sines
$\sin 260 ^ { \circ } \cos 20 ^ { \circ } - \cos 260 ^ { \circ } \sin 20 ^ { \circ }$

A) $- \frac { \sqrt { 3 } } { 2 }$
B) $- \frac { 1 } { 2 }$
C) $\frac { 13 } { 3 }$
D) $\frac { \sqrt { 3 } } { 2 }$

Question 56

Use the Formula for the Cosine of the Difference of Two Angles
$\cos \left( \frac { 5 \pi } { 12 } \right) \cos \left( \frac { \pi } { 4 } \right) + \sin \left( \frac { 5 \pi } { 12 } \right) \sin \left( \frac { \pi } { 4 } \right)$

A) $\cos \left( \frac { \pi } { 6 } \right)$
B) $\cos \left( \frac { \pi } { 3 } \right)$
C) $\cos \left( \frac { 2 \pi } { 3 } \right)$
D) $\cos \left( \frac { 5 \pi } { 6 } \right)$ Find the exact value of the expression.

Question 57

Complete the identity.
$\cos \left( \frac { 3 \pi } { 2 } - x \right) =$ ?

A) $- \sin x$
B) $\sin x$
C) $- \cos x$
D) $\cos x$

Question 58

Use the Formula for the Cosine of the Difference of Two Angles
$\cos \left( 165 ^ { \circ } \right) \cos \left( 45 ^ { \circ } \right) + \sin \left( 165 ^ { \circ } \right) \sin \left( 45 ^ { \circ } \right)$

A) $\alpha = 165 ^ { \circ } , \beta = 45 ^ { \circ }$
B) $\alpha = - 165 ^ { \circ } , \beta = 45 ^ { \circ }$
C) $\alpha = - 45 ^ { \circ } , \beta = 165 ^ { \circ }$
D) $\alpha = 45 ^ { \circ } , \beta = 165 ^ { \circ }$

Question 59

Use Sum and Difference Formulas for Cosines and Sines
$\cos 15 ^ { \circ } \cos 45 ^ { \circ } - \sin 15 ^ { \circ } \sin 45 ^ { \circ }$

A) $\frac { 1 } { 2 }$
B) $\frac { \sqrt { 3 } } { 2 }$
C) $\frac { 1 } { 4 }$
D) $\sqrt { 3 }$

Question 60

Use Sum and Difference Formulas for Cosines and Sines
$\sin \left( 190 ^ { \circ } - 70 ^ { \circ } \right)$

A) $\frac { \sqrt { 3 } } { 2 }$
B) $- \frac { 1 } { 2 }$
C) $\frac { 19 } { 6 }$
D) $- \frac { \sqrt { 3 } } { 2 }$

Question 61

Verify the identity.
$$\cos ( \alpha - \beta ) - \cos ( \alpha + \beta ) = 2 \sin \alpha \sin \beta$$

Question 62

Use the given information to find the exact value of the expression.
$\sin \alpha = - \frac { 24 } { 25 } , \alpha$ lies in quadrant IV, and $\cos \beta = - \frac { \sqrt { 21 } } { 5 } , \beta$ lies in quadrant III Find $\sin ( \alpha - \beta )$ .

A) $\frac { 14 + 24 \sqrt { 21 } } { 125 }$
B) $\frac { - 14 + 24 \sqrt { 21 } } { 125 }$
C) $\frac { - 48 + 7 \sqrt { 21 } } { 125 }$
D) $\frac { - 48 - 7 \sqrt { 21 } } { 125 }$

Question 63

Verify the identity.
$$\sin \left( \frac { 3 \pi } { 2 } - \theta \right) = - \cos \theta$$

Question 64

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

Question 65

Use Sum and Difference Formulas for Tangents
Find the exact value by using a difference identity.
$\tan 255 ^ { \circ }$

A) $\sqrt { 3 } + 2$
B) $- \sqrt { 3 } + 2$
C) $- \sqrt { 3 } - 2$
D) $\sqrt { 3 } - 2$

Question 66

Use Sum and Difference Formulas for Tangents
Find the exact value by using a difference identity.
$\tan 285 ^ { \circ }$

A) $- 2 - \sqrt { 3 }$
B) $2 + \sqrt { 3 }$
C) $\frac { 2 - \sqrt { 3 } } { 4 }$
D) $\frac { 2 + \sqrt { 3 } } { 4 }$

Question 67

Describe the graph using another equation.
$y=\sin (\pi-x)$

A) $\sin x$
B) $\cos x$
C) $- \sin x$
D) $- \cos x$

Question 68

Use the given information to find the exact value of the expression.
$\tan \alpha = \frac { 3 } { 4 } , \alpha$ lies in quadrant III, and $\cos \beta = - \frac { 12 } { 13 } , \beta$ lies in quadrant II $\quad$ Find $\sin ( \alpha + \beta )$ .

A) $\frac { 16 } { 65 }$
B) $\frac { 63 } { 65 }$
C) $\frac { 33 } { 65 }$
D) $\frac { 56 } { 65 }$

Question 69

Complete the identity.
$\frac { \sin ( \alpha + \beta ) } { \cos \alpha \cos \beta } = ?$

A) $\tan \alpha + \tan \beta$
B) $\tan \beta + \tan \alpha$
C) $\cot \alpha + \cot \beta$
D) $- \tan \alpha + \cot \beta$

Question 70

Complete the identity.
$\sin ( \alpha + \beta ) \sin ( \alpha - \beta ) =$ ?

A) $\cos ^ { 2 } \beta - \cos ^ { 2 } \alpha$
B) $\sin ^ { 2 } \alpha - \cos ^ { 2 } \beta$
C) $\cos ^ { 2 } \beta + \cos ^ { 2 } \alpha$
D) $\sin ^ { 2 } \beta - \sin ^ { 2 } \alpha$

Question 71

Verify the identity.
$$\sin ( \alpha + \beta ) - \sin ( \alpha - \beta ) = 2 \cos \alpha \sin \beta$$

Question 72

Use the given information to find the exact value of the expression.
$\cos \alpha = - \frac { 4 } { 5 } , \alpha$ lies in quadrant III, and $\sin \beta = \frac { \sqrt { 21 } } { 5 } , \beta$ lies in quadrant II $\quad$ Find $\cos ( \alpha + \beta )$

A) $\frac { 8 + 3 \sqrt { 21 } } { 25 }$
B) $\frac { - 8 - 3 \sqrt { 21 } } { 25 }$
C) $\frac { 6 - 4 \sqrt { 21 } } { 25 }$
D) $\frac { - 6 + 4 \sqrt { 21 } } { 25 }$

Question 73

Use the given information to find the exact value of the expression.
$\sin \alpha = \frac { 3 } { 5 } , \alpha$ lies in quadrant II, and $\cos \beta = \frac { 2 } { 5 } , \beta$ lies in quadrant I $\quad$ Find $\cos ( \alpha - \beta )$ .

A) $\frac { - 8 + 3 \sqrt { 21 } } { 25 }$
B) $\frac { 6 + 4 \sqrt { 21 } } { 25 }$
C) $\frac { 6 - 4 \sqrt { 21 } } { 25 }$
D) $\frac { 8 - 3 \sqrt { 21 } } { 25 }$

Question 74

Complete the identity.
$\frac { \sin ( \alpha + \beta ) } { \sin ( \alpha - \beta ) } =$ ?

A) $\frac { \tan \alpha + \tan \beta } { \tan \alpha - \tan \beta }$
B) $\frac { \tan \alpha - \tan \beta } { \tan \alpha + \tan \beta }$
C) $\frac { \tan ( \alpha + \beta ) } { \tan \alpha - \tan \beta }$
D) $\frac { \tan \alpha - \tan \beta } { \tan ( \alpha + \beta ) }$

Question 75

Describe the graph using another equation.
$y=\cos \left(x+\frac{\pi}{2}\right)-\cos \left(x-\frac{\pi}{2}\right)$

A) $- 2 \sin x$
B) $- 2 \cos x$
C) $- \sin x$
D) $- \cos x$

Question 76

Use the given information to find the exact value of the expression.
$\sin \alpha = \frac { 8 } { 17 } , \alpha$ lies in quadrant II, and $\cos \beta = \frac { 12 } { 13 } , \beta$ lies in quadrant I $\quad$ Find $\sin ( \alpha - \beta )$ .

A) $\frac { 171 } { 221 }$
B) $\frac { 220 } { 221 }$
C) $\frac { 140 } { 221 }$
D) $\frac { 21 } { 221 }$

Question 77

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

Question 78

Verify the identity.
$$\cos \left( \frac { 3 \pi } { 2 } - \theta \right) = - \sin \theta$$

Question 79

Use the given information to find the exact value of the expression.
$\sin \alpha = \frac { 20 } { 29 } , \alpha$ lies in quadrant $\mathrm { I }$ , and $\cos \beta = \frac { 4 } { 5 } , \beta$ lies in quadrant I $\quad$ Find $\cos ( \alpha + \beta )$

A) $\frac { 24 } { 145 }$
B) $\frac { 144 } { 145 }$
C) $\frac { 17 } { 145 }$
D) $\frac { 143 } { 145 }$

Question 80

Verify the identity.
$$\frac { \cos ( \alpha + \beta ) } { \cos \alpha \sin \beta } = \cot \beta - \tan \alpha$$