Suppose $\tan \alpha=\frac{1}{4}$ and $\tan \beta=\frac{3}{5}$, where $0<\alpha<\beta<\frac{\pi}{2}$. Find $\tan (\alpha+\beta)$. Then show that $\operatorname{Tan}^{-1} \frac{1}{4}+\operatorname{Tan}^{-1} \frac{3}{5}=\frac{\pi}{4}$.

Find the exact value of the given expression.

$1-2 \sin ^{2} \frac{7 \pi}{12}$.

Solve each inequality for $0 \leq x<2 \pi$. Give answers to the nearest hundredth. You may wish to use a graphing calculator or computer.

$\cos 2 x \geq 5-\cos x$.

Find the exact value of the given expression.

$\cos ^{2} \frac{\pi}{12}-\sin ^{2} \frac{\pi}{12}$.

Solve each inequality for $0 \leq x<2 \pi$. Give answers to the nearest hundredth. You may wish to use a graphing calculator or computer.

$$\cos x \leq \sin 2 x$$

Suppose that sin $\alpha=\frac{4}{5}$ and sin $\beta=\frac{1}{2}$, where $\frac{\pi}{2}<\beta<\alpha<\pi$. Find sin ($\alpha-\beta)$

Use a graphing calculator set in radian mode to sketch each graph. Explain how the graph is related to the graph of v = tan x and why.

$$y=\frac{\tan x+\tan 1}{1-\tan x \tan 1}$$

Find the exact value of the expression.

cos 105 degrees

Solve each inequality for $0 \leq x<2 \pi$. Give answers to the nearest hundredth. You may wish to use a graphing calculator or computer.

$2 \cos x \leq \cos \left(x-\frac{\pi}{2}\right)$.

Find the exact value of the given expression.

$\frac{2 \tan \frac{\pi}{8}}{1-\tan ^{2} \frac{\pi}{8}}$.

Find the two supplementary angles formed by the line 3x + 2y = 5 and the line 4x - 3y = 1

Find the exact value of the expression.

cos 15 degrees

Find the exact value of the expression.

cos 75 degrees

Find the exact value of the given expression.

$2 \cos ^{2} \frac{\pi}{8}-1$.