91Ó°ÊÓ

Simplify each expression using half-angle identities. Do not evaluate. $$ \sqrt{\frac{1-\cos \left(\frac{\pi}{4}\right)}{2}} $$

In Exercises 31-44, verify the identities. $$ \sin ^{2}\left(\frac{x}{2}\right)+\cos ^{2}\left(\frac{x}{2}\right)=1 $$

Verify each of the trigonometric identities. $$ \frac{1}{1-\sin x}+\frac{1}{1+\sin x}=2 \sec ^{2} x $$

Write each expression as a single trigonometric function. $$ \frac{\tan 49^{\circ}+\tan 23^{\circ}}{1-\tan 49^{\circ} \tan 23^{\circ}} $$

Verify the identities. $$ 2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)=\sin x $$

Determine whether each equation is a conditional equation or an identity. $$ \cos (A+B)+\cos (A-B)=2 \cos A \cos B $$

Verify each of the trigonometric identities. $$ \tan x(\csc x-\sin x)=\cos x $$

In Exercises \(43-46\), graph the functions. $$ y=\frac{\sin (2 x)}{1-\cos (2 x)} $$

Determine whether each equation is a conditional equation or an identity. $$ \cos (2 x)=\cos ^{2} x-\sin ^{2} x $$

Graph the functions. $$ y=-6 \sin ^{2}\left(\frac{x}{2}\right) $$