Question 1

Use an addition or subtraction formula to find the solutions of the equation that are in the interval $[ 0 , \pi )$ . $\cos 8 t \cos 5 t = - \sin 8 t \sin 5 t$

Accepted Answer

A)  $t = \frac { \pi } { 6 } , t = \frac { \pi } { 2 } , t = \frac { 5 \pi } { 6 }$ 
B)  $t = \frac { 3 \pi } { 5 } , t = \frac { 4 \pi } { 5 }$ 
C)  $t = \frac { \pi } { 3 } , t = \frac { 2 \pi } { 3 }$ 
D)  $t = \frac { \pi } { 8 } , t = \frac { 3 \pi } { 8 } , t = \frac { 5 \pi } { 8 }$ 
E)  $t = 0 , t = \frac { \pi } { 3 } , t = \frac { 2 \pi } { 3 }$ 
A)  $t = \frac { \pi } { 6 } , t = \frac { \pi } { 2 } , t = \frac { 5 \pi } { 6 }$ 
B)  $t = \frac { 3 \pi } { 5 } , t = \frac { 4 \pi } { 5 }$ 
C)  $t = \frac { \pi } { 3 } , t = \frac { 2 \pi } { 3 }$ 
D)  $t = \frac { \pi } { 8 } , t = \frac { 3 \pi } { 8 } , t = \frac { 5 \pi } { 8 }$ 
E)  $t = 0 , t = \frac { \pi } { 3 } , t = \frac { 2 \pi } { 3 }$ A

Question 2

Find all solutions of the equation. $$( \cos \theta - 1 ) ( \sin \theta + 1 ) = 0$$

Accepted Answer

A)  $2 \pi$ and $- \frac { \pi } { 2 } + 2 \pi n$ for every integer n. 
B)  $2 \pi$ and $\pi + 2 \pi n$ for every integer n. 
C)  $\pi + 2 \pi n$ and $\frac { \pi } { 2 } + 2 \pi n$ for every integer n. 
D)  $\frac { \mathrm { mn } } { 2 }$ for every integer n. 
E)  $\pi + 2 \pi n$ and $- \frac { \pi } { 2 } + \pi n$ for every integer n. 
A)  $2 \pi$ and $- \frac { \pi } { 2 } + 2 \pi n$ for every integer n. 
B)  $2 \pi$ and $\pi + 2 \pi n$ for every integer n. 
C)  $\pi + 2 \pi n$ and $\frac { \pi } { 2 } + 2 \pi n$ for every integer n. 
D)  $\frac { \mathrm { mn } } { 2 }$ for every integer n. 
E)  $\pi + 2 \pi n$ and $- \frac { \pi } { 2 } + \pi n$ for every integer n. A

Question 3

Express as a sum or difference. $\sin 7 t \sin 2 t$

Accepted Answer

A)  $\cos ( 5 t ) - \cos ( 9 t )$ 
B)  $\frac { 1 } { 2 } \cos ( 6 t ) - \frac { 1 } { 2 } \cos ( 10 t )$ 
C)  $\sin ( 5 t ) - \sin ( 9 t )$ 
D)  $\frac { 1 } { 2 } \cos ( 5 t ) - \frac { 1 } { 2 } \cos ( 9 t )$ 
E)  $\frac { 1 } { 2 } \sin ( 5 t ) - \frac { 1 } { 2 } \sin ( 9 t )$ 
A)  $\cos ( 5 t ) - \cos ( 9 t )$ 
B)  $\frac { 1 } { 2 } \cos ( 6 t ) - \frac { 1 } { 2 } \cos ( 10 t )$ 
C)  $\sin ( 5 t ) - \sin ( 9 t )$ 
D)  $\frac { 1 } { 2 } \cos ( 5 t ) - \frac { 1 } { 2 } \cos ( 9 t )$ 
E)  $\frac { 1 } { 2 } \sin ( 5 t ) - \frac { 1 } { 2 } \sin ( 9 t )$ D

Question 4

Verify the identity. $\frac { \cos 4 t + \cos 5 t + \cos 6 t } { \sin 4 t + \sin 5 t + \sin 6 t } = \cot 5 t$

Accepted Answer

A)  Not identical 
B)  Identical 
A)  Not identical 
B)  Identical

Question 5

Find an identical expression. $\arcsin ( - x )$

Accepted Answer

A)  $2 \cos ^ { - 1 } x$ 
B)  $\cos ^ { - 1 } \left( 2 x ^ { 2 } - 1 \right) ; 0 \leq x \leq 1$ 
C)  $\tan ^ { - 1 } \frac { \sqrt { 1 - x ^ { 2 } } } { x }$ 
D)  $\sin ^ { - 1 } \left( 2 x ^ { 2 } - 1 \right) ; \quad 0 \leq x \leq 1$ 
E)  $- \arcsin x$ 
A)  $2 \cos ^ { - 1 } x$ 
B)  $\cos ^ { - 1 } \left( 2 x ^ { 2 } - 1 \right) ; 0 \leq x \leq 1$ 
C)  $\tan ^ { - 1 } \frac { \sqrt { 1 - x ^ { 2 } } } { x }$ 
D)  $\sin ^ { - 1 } \left( 2 x ^ { 2 } - 1 \right) ; \quad 0 \leq x \leq 1$ 
E)  $- \arcsin x$

Question 6

Express as a product. $\cos 9 \theta - \cos \theta$

Accepted Answer

A)  $- 2 \sin ( 5 \theta ) \sin ( 4 \theta )$ 
B)  $- 2 \sin ( 10 \theta ) \sin ( 8 \theta )$ 
C)  $2 \cos ( 5 \theta ) \cos ( 4 \theta )$ 
D)  $2 \cos ( 4 \theta ) \sin ( 3 \theta )$ 
E)  $- 2 \sin ( 4 \theta ) \cos ( 3 \theta )$ 
A)  $- 2 \sin ( 5 \theta ) \sin ( 4 \theta )$ 
B)  $- 2 \sin ( 10 \theta ) \sin ( 8 \theta )$ 
C)  $2 \cos ( 5 \theta ) \cos ( 4 \theta )$ 
D)  $2 \cos ( 4 \theta ) \sin ( 3 \theta )$ 
E)  $- 2 \sin ( 4 \theta ) \cos ( 3 \theta )$

Question 7

Use the formula $a \cos B x + b \sin B x = A \cos ( B x - C )$ $\text { where } A = \sqrt { a ^ { 2 } + b ^ { 2 } } \text { and } \tan C = \frac { b } { a } \text { with } - \frac { \pi } { 2 } < C < \frac { \pi } { 2 }$ to determine the period of f (x). $f ( x ) = 7 \cos 12 x - 7 \sin 12 x$

Accepted Answer

A)  $\frac { \pi } { 7 }$ 
B)  $\frac { \pi } { 6 }$ 
C)  $\frac { \pi } { 4 }$ 
D)  $6 \pi$ 
E)  $\frac { \pi } { 12 }$ 
A)  $\frac { \pi } { 7 }$ 
B)  $\frac { \pi } { 6 }$ 
C)  $\frac { \pi } { 4 }$ 
D)  $6 \pi$ 
E)  $\frac { \pi } { 12 }$

Question 8

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

Accepted Answer

A)  invalid 
B)  valid 
A)  invalid 
B)  valid

Question 9

Find the exact values of $\sin 2 \theta , \cos 2 \theta , \text { and } \tan 2 \theta$ for the given values of $6$ . $\sec \theta = - 12 ; 90 ^ { \circ } < \theta < 180 ^ { \circ }$

Accepted Answer

A)  $- \frac { \sqrt { 143 } } { 72 } , - \frac { 71 } { 72 } , \frac { \sqrt { 143 } } { 71 }$ 
B)  $\frac { \sqrt { 12 } } { 143 } , \frac { 71 } { 72 } , \frac { \sqrt { 71 } } { 143 }$ 
C)  $- \frac { 143 } { 72 } , - \frac { 71 } { 72 } , - \frac { 143 } { 71 }$ 
D)  $- \frac { \sqrt { 143 } } { 72 } , - \frac { 143 } { 72 } , \frac { \sqrt { 143 } } { 143 }$ 
E)  $- \frac { 143 } { 72 } , \frac { 71 } { 72 } , \frac { 143 } { 71 }$ 
A)  $- \frac { \sqrt { 143 } } { 72 } , - \frac { 71 } { 72 } , \frac { \sqrt { 143 } } { 71 }$ 
B)  $\frac { \sqrt { 12 } } { 143 } , \frac { 71 } { 72 } , \frac { \sqrt { 71 } } { 143 }$ 
C)  $- \frac { 143 } { 72 } , - \frac { 71 } { 72 } , - \frac { 143 } { 71 }$ 
D)  $- \frac { \sqrt { 143 } } { 72 } , - \frac { 143 } { 72 } , \frac { \sqrt { 143 } } { 143 }$ 
E)  $- \frac { 143 } { 72 } , \frac { 71 } { 72 } , \frac { 143 } { 71 }$

Question 10

Find the solutions of the equation that are in the interval $[ 0,2 \pi ) .$ $\cot ^ { 2 } u \cos u = \cos u$

Accepted Answer

A)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
B)  $0 , \pi , \frac { \pi } { 2 } , \frac { 3 \pi } { 2 }$ 
C)  $0 , \pi , \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
D)  $\frac { \pi } { 2 } , \frac { 3 \pi } { 2 } , \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
E)  no solution 
A)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
B)  $0 , \pi , \frac { \pi } { 2 } , \frac { 3 \pi } { 2 }$ 
C)  $0 , \pi , \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
D)  $\frac { \pi } { 2 } , \frac { 3 \pi } { 2 } , \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
E)  no solution

Question 11

Express as a product. $\sin 13 \theta + \sin 3 \theta$

Accepted Answer

A)  $\sin ( 8 \theta ) \cos ( 5 \theta )$ 
B)  $2 \sin ( 16 \theta ) \cos ( 10 \theta )$ 
C)  $2 \cos ( 9 \theta ) \sin ( 4 \theta )$ 
D)  $4 \sin ( 8 \theta ) \cos ( 5 \theta )$ 
E)  $2 \sin ( 8 \theta ) \cos ( 5 \theta )$ 
A)  $\sin ( 8 \theta ) \cos ( 5 \theta )$ 
B)  $2 \sin ( 16 \theta ) \cos ( 10 \theta )$ 
C)  $2 \cos ( 9 \theta ) \sin ( 4 \theta )$ 
D)  $4 \sin ( 8 \theta ) \cos ( 5 \theta )$ 
E)  $2 \sin ( 8 \theta ) \cos ( 5 \theta )$

Question 12

Find all solutions of the equation. $$\tan \theta = - \frac { 1 } { \sqrt { 3 } }$$

Accepted Answer

A)  $\frac { 5 \pi } { 6 } + \pi n$ for every integer n. 
B)  $\frac { 5 \pi } { 6 } + 2 \pi n$ for every integer n. 
C)  $\frac { \pi } { 3 } + \pi n$ for every integer n. 
D)  $\frac { 2 \pi } { 3 } + \frac { \pi n } { 2 }$ for every integer n. 
E)  $\frac { \pi } { 6 } + 2 \pi n$ for every integer n. 
A)  $\frac { 5 \pi } { 6 } + \pi n$ for every integer n. 
B)  $\frac { 5 \pi } { 6 } + 2 \pi n$ for every integer n. 
C)  $\frac { \pi } { 3 } + \pi n$ for every integer n. 
D)  $\frac { 2 \pi } { 3 } + \frac { \pi n } { 2 }$ for every integer n. 
E)  $\frac { \pi } { 6 } + 2 \pi n$ for every integer n.

Question 13

Verify the identity. $$\ln \sec \theta = - \ln \cos \theta$$

Accepted Answer

A)  valid 
B)  invalid 
A)  valid 
B)  invalid

Question 14

Verify the identity. $\frac { \sin \theta + \sin 9 \theta } { \cos \theta + \cos 9 \theta } = \tan 9 \theta$

Accepted Answer

A)  Not identical 
B)  Identical 
A)  Not identical 
B)  Identical

Question 15

Use the formula $a \cos B x + b \sin B x = A \cos ( B x - C )$ $\text { where } A = \sqrt { a ^ { 2 } + b ^ { 2 } } \text { and } \tan C = \frac { b } { a } \text { with } - \frac { \pi } { 2 } < C < \frac { \pi } { 2 }$ to determine the amplitude of f (x). $f ( x ) = 2 \cos 4 x - 2 \sin 4 x$

Accepted Answer

A)  $A = 11 \sqrt { 2 }$ 
B)  $A = 9 \sqrt { 2 }$ 
C)  $A = 2 \sqrt { 6 }$ 
D)  $A = 2 \sqrt { 2 }$ 
E)  $A = 2$ 
A)  $A = 11 \sqrt { 2 }$ 
B)  $A = 9 \sqrt { 2 }$ 
C)  $A = 2 \sqrt { 6 }$ 
D)  $A = 2 \sqrt { 2 }$ 
E)  $A = 2$

Question 16

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

Accepted Answer

A)  valid 
B)  invalid 
A)  valid 
B)  invalid

Question 17

A golfer, centered in a b = 40-yard-wide straight fairway, hits a ball a = 230 yards. Approximate the largest angle the drive can have from the center of the fairway if the ball is to stay in the fairway (see the figure). Round the answer to the nearest hundredth.

Accepted Answer

A)  $4.67 ^ { \circ }$ 
B)  $4.97 ^ { \circ }$ 
C)  $10 ^ { \circ }$ 
D)  $5.17 ^ { \circ }$ 
E)  $5.7 ^ { \circ }$ 
A)  $4.67 ^ { \circ }$ 
B)  $4.97 ^ { \circ }$ 
C)  $10 ^ { \circ }$ 
D)  $5.17 ^ { \circ }$ 
E)  $5.7 ^ { \circ }$

Question 18

Verify the identity. $4 \cos 3 x \cos 6 x \sin 9 x = \sin 3 x + \sin 14 x + \sin 19 x$

Accepted Answer

A)  Not identical 
B)  Identical 
A)  Not identical 
B)  Identical

Question 19

Use the graph of f to find the simplest expression g(x) such that the equation f (x) = g (x) is an identity. $f ( x ) = \frac { \sin 4 t + \sin 5 t + \sin 6 t } { \cos 4 t + \cos 5 t + \cos 6 t }$

Accepted Answer

A)  $g ( x ) = \cot ( 5 x )$ 
B)  $g ( x ) = \tan ( 5 x )$ 
C)  $g ( x ) = \cot ( 6 x )$ 
D)  $g ( x ) = \tan ( 6 x )$ 
E)  $g ( x ) = \tan ( 4 x )$ 
A)  $g ( x ) = \cot ( 5 x )$ 
B)  $g ( x ) = \tan ( 5 x )$ 
C)  $g ( x ) = \cot ( 6 x )$ 
D)  $g ( x ) = \tan ( 6 x )$ 
E)  $g ( x ) = \tan ( 4 x )$

Question 20

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

Accepted Answer

A)  valid 
B)  invalid 
A)  valid 
B)  invalid

Use an addition or subtraction formula to find the solutions of the equation that are in the interval $[ 0 , \pi )$ . $\cos 8 t \cos 5 t = - \sin 8 t \sin 5 t$

Find all solutions of the equation. $( \cos \theta - 1 ) ( \sin \theta + 1 ) = 0$

Express as a sum or difference. $\sin 7 t \sin 2 t$

Verify the identity. $\frac { \cos 4 t + \cos 5 t + \cos 6 t } { \sin 4 t + \sin 5 t + \sin 6 t } = \cot 5 t$

Find an identical expression. $\arcsin ( - x )$

Express as a product. $\cos 9 \theta - \cos \theta$

Use the formula $a \cos B x + b \sin B x = A \cos ( B x - C )$ $\text { where } A = \sqrt { a ^ { 2 } + b ^ { 2 } } \text { and } \tan C = \frac { b } { a } \text { with } - \frac { \pi } { 2 } < C < \frac { \pi } { 2 }$ to determine the period of f (x). $f ( x ) = 7 \cos 12 x - 7 \sin 12 x$

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

Find the exact values of $\sin 2 \theta , \cos 2 \theta , \text { and } \tan 2 \theta$ for the given values of $6$ . $\sec \theta = - 12 ; 90 ^ { \circ } < \theta < 180 ^ { \circ }$

Find the solutions of the equation that are in the interval $[ 0,2 \pi ) .$ $\cot ^ { 2 } u \cos u = \cos u$

Express as a product. $\sin 13 \theta + \sin 3 \theta$

Find all solutions of the equation. $\tan \theta = - \frac { 1 } { \sqrt { 3 } }$

Verify the identity. $\ln \sec \theta = - \ln \cos \theta$

Verify the identity. $\frac { \sin \theta + \sin 9 \theta } { \cos \theta + \cos 9 \theta } = \tan 9 \theta$

Use the formula $a \cos B x + b \sin B x = A \cos ( B x - C )$ $\text { where } A = \sqrt { a ^ { 2 } + b ^ { 2 } } \text { and } \tan C = \frac { b } { a } \text { with } - \frac { \pi } { 2 } < C < \frac { \pi } { 2 }$ to determine the amplitude of f (x). $f ( x ) = 2 \cos 4 x - 2 \sin 4 x$

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

A golfer, centered in a b = 40-yard-wide straight fairway, hits a ball a = 230 yards. Approximate the largest angle the drive can have from the center of the fairway if the ball is to stay in the fairway (see the figure). Round the answer to the nearest hundredth.

Verify the identity. $4 \cos 3 x \cos 6 x \sin 9 x = \sin 3 x + \sin 14 x + \sin 19 x$

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

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Exam 6: Analytic Trigonometry

Use an addition or subtraction formula to find the solutions of the equation that are in the interval [0,π)[ 0 , \pi )[0,π) . cos⁡8tcos⁡5t=−sin⁡8tsin⁡5t\cos 8 t \cos 5 t = - \sin 8 t \sin 5 tcos8tcos5t=−sin8tsin5t

Find all solutions of the equation. (cos⁡θ−1)(sin⁡θ+1)=0( \cos \theta - 1 ) ( \sin \theta + 1 ) = 0(cosθ−1)(sinθ+1)=0

Express as a sum or difference. sin⁡7tsin⁡2t\sin 7 t \sin 2 tsin7tsin2t

Verify the identity. cos⁡4t+cos⁡5t+cos⁡6tsin⁡4t+sin⁡5t+sin⁡6t=cot⁡5t\frac { \cos 4 t + \cos 5 t + \cos 6 t } { \sin 4 t + \sin 5 t + \sin 6 t } = \cot 5 tsin4t+sin5t+sin6tcos4t+cos5t+cos6t​=cot5t

Find an identical expression. arcsin⁡(−x)\arcsin ( - x )arcsin(−x)

Express as a product. cos⁡9θ−cos⁡θ\cos 9 \theta - \cos \thetacos9θ−cosθ

Verify the identity. sin⁡x+cos⁡xcot⁡x=csc⁡x\sin x + \cos x \cot x = \csc xsinx+cosxcotx=cscx

Find the exact values of sin⁡2θ,cos⁡2θ, and tan⁡2θ\sin 2 \theta , \cos 2 \theta , \text { and } \tan 2 \thetasin2θ,cos2θ, and tan2θ for the given values of 666 . sec⁡θ=−12;90∘<θ<180∘\sec \theta = - 12 ; 90 ^ { \circ } < \theta < 180 ^ { \circ }secθ=−12;90∘<θ<180∘

Find the solutions of the equation that are in the interval [0,2π).[ 0,2 \pi ) .[0,2π). cot⁡2ucos⁡u=cos⁡u\cot ^ { 2 } u \cos u = \cos ucot2ucosu=cosu

Express as a product. sin⁡13θ+sin⁡3θ\sin 13 \theta + \sin 3 \thetasin13θ+sin3θ

Find all solutions of the equation. tan⁡θ=−13\tan \theta = - \frac { 1 } { \sqrt { 3 } }tanθ=−3​1​

Verify the identity. ln⁡sec⁡θ=−ln⁡cos⁡θ\ln \sec \theta = - \ln \cos \thetalnsecθ=−lncosθ

Verify the identity. sin⁡θ+sin⁡9θcos⁡θ+cos⁡9θ=tan⁡9θ\frac { \sin \theta + \sin 9 \theta } { \cos \theta + \cos 9 \theta } = \tan 9 \thetacosθ+cos9θsinθ+sin9θ​=tan9θ

Verify the identity. sin⁡x+cos⁡xcot⁡x=csc⁡x\sin x + \cos x \cot x = \csc xsinx+cosxcotx=cscx

A golfer, centered in a b = 40-yard-wide straight fairway, hits a ball a = 230 yards. Approximate the largest angle the drive can have from the center of the fairway if the ball is to stay in the fairway (see the figure). Round the answer to the nearest hundredth.

Verify the identity. 4cos⁡3xcos⁡6xsin⁡9x=sin⁡3x+sin⁡14x+sin⁡19x4 \cos 3 x \cos 6 x \sin 9 x = \sin 3 x + \sin 14 x + \sin 19 x4cos3xcos6xsin9x=sin3x+sin14x+sin19x

Use the graph of f to find the simplest expression g(x) such that the equation f (x) = g (x) is an identity. f(x)=sin⁡4t+sin⁡5t+sin⁡6tcos⁡4t+cos⁡5t+cos⁡6tf ( x ) = \frac { \sin 4 t + \sin 5 t + \sin 6 t } { \cos 4 t + \cos 5 t + \cos 6 t }f(x)=cos4t+cos5t+cos6tsin4t+sin5t+sin6t​

Verify the identity. tan⁡2xsec⁡x+1=1−cos⁡xcos⁡x\frac { \tan ^ { 2 } x } { \sec x + 1 } = \frac { 1 - \cos x } { \cos x }secx+1tan2x​=cosx1−cosx​

Topics From Algebra

Functions and Graphs

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

Applications of Trigonometry

Systems of Equations and Inequalities

Sequences, Series, and Probability

Topics From Analytical Geometry

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Use an addition or subtraction formula to find the solutions of the equation that are in the interval $[ 0 , \pi )$ . $\cos 8 t \cos 5 t = - \sin 8 t \sin 5 t$

Find all solutions of the equation. $( \cos \theta - 1 ) ( \sin \theta + 1 ) = 0$

Express as a sum or difference. $\sin 7 t \sin 2 t$

Verify the identity. $\frac { \cos 4 t + \cos 5 t + \cos 6 t } { \sin 4 t + \sin 5 t + \sin 6 t } = \cot 5 t$

Find an identical expression. $\arcsin ( - x )$

Express as a product. $\cos 9 \theta - \cos \theta$

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

Find the exact values of $\sin 2 \theta , \cos 2 \theta , \text { and } \tan 2 \theta$ for the given values of $6$ . $\sec \theta = - 12 ; 90 ^ { \circ } < \theta < 180 ^ { \circ }$

Find the solutions of the equation that are in the interval $[ 0,2 \pi ) .$ $\cot ^ { 2 } u \cos u = \cos u$

Express as a product. $\sin 13 \theta + \sin 3 \theta$

Find all solutions of the equation. $\tan \theta = - \frac { 1 } { \sqrt { 3 } }$

Verify the identity. $\ln \sec \theta = - \ln \cos \theta$

Verify the identity. $\frac { \sin \theta + \sin 9 \theta } { \cos \theta + \cos 9 \theta } = \tan 9 \theta$

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

Verify the identity. $4 \cos 3 x \cos 6 x \sin 9 x = \sin 3 x + \sin 14 x + \sin 19 x$

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