91Ó°ÊÓ

Write each expression as an equivalent expression involving only \(x\). (Assume \(x\) is positive.) $$ \sin \left(2 \cos ^{-1} x\right) $$

Graph each of the following from \(x=0\) to \(x=2 \pi\). \(y=4 \cos ^{2} x-2\)

Use exact values to show that each of the following is true. \(\sin 90^{\circ}=2 \sin 45^{\circ} \cos 45^{\circ}\)

Graph each of the following from \(x=0\) to \(x=2 \pi\). $$y=\sin x \cos 2 x+\cos x \sin 2 x$$

Graph one complete cycle of \(y=\sin x \cos \frac{\pi}{6}-\cos x \sin \frac{\pi}{6}\) by first rewriting the right side in the form \(\sin (A-B)\).

If \(\sin A=\frac{4}{5}\) with \(A\) in \(\mathrm{QII}\), and \(\sin B=\frac{3}{5}\) with \(B\) in QI, find \(\cos (A-B)\)

Write a formula for \(\sin 2 x\) by writing \(\sin 2 x\) as \(\sin (x+x)\) and using the formula for the sine of a sum.

Prove the following identities. \(\sec ^{2} \frac{A}{2}=\frac{2 \sec A}{\sec A+1}\)

Prove each of the following identities. \(\frac{1-\tan x}{1+\tan x}=\frac{1-\sin 2 x}{\cos 2 x}\)

Use your graphing calculator to determine if each equation appears to be an identity by graphing the left expression and right expression together. If so, prove the identity. If not, find a counterexample. $$\sin x=\cos \left(\frac{\pi}{2}-x\right)$$