Find:

(a) $\tan (\alpha+\beta)$ and

(b) $\tan (\alpha-\beta)$

(c) $\tan \alpha=2$

(d) $\tan \beta=-\frac{1}{3}$

Solve the equations for $0^{\circ} \leq x<360^{\circ}$.

$\sin ^{2} x=\sin x$.

Simplify the given expression.

$2 \sin \frac{\alpha}{2} \cos \frac{\alpha}{2}$.

Suppose $\alpha=\operatorname{Tan}^{-1} 2$ and $\beta=\operatorname{Tan}^{-1} 3$. Show that $\tan (\alpha+\beta)=-1$.

Angle A is acute. If If $\tan A=\frac{1}{2}$, find cos 2 A and tan 2 A.

Solve each equation for $0^{\circ} \leq x<360^{\circ}$ by using trigonometric identities. Give answers to the nearest tenth when necessary. You may wish to check your answers using a graphing calculator or computer.

$\sin \left(60^{\circ}-x\right)=2 \sin x$.

Find the exact value of the given expression.

$4 \sin \frac{2 \pi}{3} \cos \frac{2 \pi}{3}$.

Suppose $\tan \alpha=3$ and $\tan \beta=\frac{1}{2}$. where $0<\beta<\alpha<\frac{\pi}{2}$. Find $\tan (\alpha-\beta)$.

Solve each equation for $0^{\circ} \leq x<360^{\circ}$ by using trigonometric identities. Give answers to the nearest tenth when necessary. You may wish to check your answers using a graphing calculator or computer.

$\cot \left(x-20^{\circ}\right)=1$.

Find the exact value of the expression.

cos (-165 degrees)

Find the exact value of the expression.

sin (-15 degrees)

Solve each equation for $0^{\circ} \leq x<360^{\circ}$ by using trigonometric identities. Give answers to the nearest tenth when necessary. You may wish to check your answers using a graphing calculator or computer.

$2 \cos \left(x+45^{\circ}\right)=1$.

Find the exact value of the given expression.

$\sin 15^{\circ} \cos 15^{\circ}$.

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 $\operatorname{Tan}^{-1} \frac{1}{4}+\operatorname{Tan}^{-1} \frac{3}{5}=\frac{\pi}{4}$.