Exam 7: Analytic Trigonometry

Use a double-angle formula or a half-angle formula to simplify $\frac { 2 \tan 32 } { 1 - \tan ^ { 2 } 32 }$ .

Solve the given equation. $\cos \theta = - 1$

Show that the equation $\cos ( x - y ) = \cos x - \cos y$ is not an identity.

Find all solutions of the given equation. $4 \sin \theta - 1 = 0$

Solve the given equation. $\sin \theta = 0.25$

Use an addition or subtraction formula to write $\sin 23 ^ { \circ } \cos 22 ^ { \circ } + \cos 23 ^ { \circ } \sin 22 ^ { \circ }$ and find its exact value.

Find all solutions of $2 \sin x - \sqrt { 2 } = 0$ .

Solve the given equation. $\tan \theta = - \frac { 1 } { 2 }$

Prove $\cot \left( x + \frac { \pi } { 4 } \right) = \frac { 1 - \tan x } { \tan x + 1 }$ .

Write $\cos x + \cos 5 x$ as a product.

Verify the identity. $\frac { 1 + \cos x } { 1 - \cos x } - \frac { 1 - \cos x } { 1 + \cos x } = 4 \cot x \csc x$

Prove $\tan \left( x + \frac { \pi } { 4 } \right) = \frac { 1 + \tan x } { 1 - \tan x }$ .

Solve the given equation. $4 \sin ^ { 2 } \theta - 4 \sin \theta + 1 = 0$

Use a double-angle formula or half-angle formula to simplify $\cos ^ { 2 } \frac { \pi } { 8 } - \sin ^ { 2 } \frac { \pi } { 8 }$ .

Use addition or subtraction formulas to find the exact value of $\cos \left( 255 ^ { \circ } \right)$ .

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

Simplify. $\frac { \cot x } { \csc ( - x ) }$

Find all solutions of $\tan ^ { 2 } x + ( \sqrt { 3 } - 1 ) \tan x - \sqrt { 3 } = 0$ .

Find all solutions of $\tan x \cos x - \cos x = 0$ .