91Ó°ÊÓ

Find the exact value of \(\sin (\alpha-\beta)\) if \(\sin \alpha=-\frac{3}{5}\) and \(\sin \beta=\frac{1}{5}\), if the terminal side of \(\alpha\) lies in quadrant III and the terminal side of \(\beta\) lies in quadrant I.

Verify each identity. $$ \frac{1}{2} \sin (4 x)=2 \sin x \cos x-4 \sin ^{3} x \cos x $$

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

Verify the identities. $$ \frac{\cos ^{2} A-\cos ^{2} B}{\cos A+\cos B}=-2\left[\sin \left(\frac{A}{2}\right) \cos \left(\frac{B}{2}\right)+\cos \left(\frac{A}{2}\right) \sin \left(\frac{B}{2}\right)\right]\left[\sin \left(\frac{A}{2}\right) \cos \left(\frac{B}{2}\right)-\cos \left(\frac{A}{2}\right) \sin \left(\frac{B}{2}\right)\right] $$

Verify each identity. $$ \cos (4 x)=[\cos (2 x)-\sin (2 x)][\cos (2 x)+\sin (2 x)] $$

Verify the identities. $$ \sin A=\sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)+\sin \left(\frac{A-B}{2}\right) \cos \left(\frac{A+B}{2}\right) $$

Verify each of the trigonometric identities. $$ \frac{\sec x+\tan x}{\csc x+1}=\tan x $$

Find the exact value of \(\sin (\alpha+\beta)\) if \(\sin \alpha=-\frac{3}{5}\) and \(\sin \beta=\frac{1}{5}\), if the terminal side of \(\alpha\) lies in quadrant III and the termi side of \(\beta\) lies in quadrant II.

Verify the identities. $$ \tan ^{2}\left(\frac{x}{2}\right)=(\csc x-\cot x)^{2} $$

Verify the identities. $$ \tan \left(\frac{A}{2}\right)+\cot \left(\frac{A}{2}\right)=2 \csc A $$