Exam 5: Analytic Trigonometry

Determine which of the following are trigonometric identities. 3 $\text { I. } \frac { \cos ( 4 x ) - \cos ( 2 x ) } { 2 \tan ( 3 x ) } = - \tan ( x )$ II. $\frac { \cos ( 3 x ) - \cos ( x ) } { \sin ( 3 x ) - \sin ( x ) } = - \tan ( 2 x )$ $\frac { \cos ( 6 x ) - \cos ( 2 x ) } { \sin ( 4 x ) + \sin ( 2 x ) } = - \tan ( 3 x$

Verify the identity shown below. $\sec ^ { 2 } \mu - \cot ^ { 2 } \left( \frac { \pi } { 2 } - \mu \right) = 1$

Simplify the given expression algebraically. $\cos ( \pi + x )$

Verify the identity shown below. $\frac { \tan \theta + 1 } { \sec \theta + \csc \theta } = \sin \theta$

Find all solutions of the following equation in the interval $[ 0,2 \pi )$ . $\csc ^ { 2 } x = \cot x + 1$

Solve the following equation. $\csc ^ { 2 } ( x ) - 4 = 0$

Find the exact value of $\cos ( u - v )$ given that $\sin u = - \frac { 9 } { 41 }$ and $\cos v = \frac { 15 } { 17 }$ . (Both $u$ and $v$ are in Quadrant IV.)

Verify the given identity. $\frac { \sin u + \sin v } { \cos u + \cos v } = \tan \frac { 1 } { 2 } ( u + v )$

Rewrite the expression $\frac { \sin ( y ) } { 1 - \cos ( y ) }$ so that it is not in fractional form.

Use the cofunction identities to evaluate the expression below without the aid of a calculator. $\cos ^ { 2 } 50 ^ { \circ } + \cos ^ { 2 } 58 ^ { \circ } + \cos ^ { 2 } 40 ^ { \circ } + \cos ^ { 2 } 32 ^ { \circ }$

Use the figure below to find the exact value of the given trigonometric expression $\cot \frac { x } { 2 }$

Which of the following is a solution to the given equation? $\sec x - 2 = 0$

Approximate the solutions of the equation $2 \sin ^ { 2 } ( x ) + 5 \sin ( x ) - 2 = 0$ by considering its graph below. Round your answer to one decimal.

Solve the following equation. $\tan ^ { 2 } x + \tan x = 0$

Verify the given identity. $\frac { \sin 5 x + \sin 3 x } { \cos 5 x + \cos 3 x } = \tan 4 x$

Use the half-angle formula to simplify the given expression. $\sqrt { \frac { 1 + \cos 16 x } { 2 } }$

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $\tan ^ { 3 } x - \tan ^ { 2 } x + \tan x - 1$

Determine which of the following are trigonometric identities. I. $\frac { \sin ( x ) - \sin ( y ) } { \cos ( x ) + \cos ( y ) } + \frac { \cos ( x ) - \cos ( y ) } { \sin ( x ) + \sin ( y ) } = 0$ II. $\frac { \sin ( x ) + \sin ( y ) } { \cos ( x ) + \cos ( y ) } + \frac { \cos ( x ) + \cos ( y ) } { \sin ( x ) + \sin ( y ) } = 1$ III. $\frac { \sin ( x ) + \cos ( y ) } { \sin ( x ) \cos ( y ) } = \sin ( y ) + \cos ( x )$

Solve the multi-angle equation below. $\cos \left( \frac { x } { 2 } \right) = \frac { \sqrt { 2 } } { 2 }$

Verify the identity shown below. $\frac { 1 + \tan \theta } { 1 + \cot \theta } = \tan \theta$