91Ó°ÊÓ

Find the exact value of the following under the given conditions: a. \(\cos (\alpha+\beta)\) b. \(\sin (\alpha+\beta)\) c. \(\tan (\alpha+\beta)\) \(\tan \alpha=-\frac{4}{3}, \alpha\) lies in quadrant II, and \(\cos \beta=\frac{2}{3}, \beta\) lies in quadrant I.

Solve each equation on the interval \([0,2 \pi)\) $$ \cos x-2 \sin x \cos x=0 $$

Verify each identity. $$ \sin ^{2} \frac{\theta}{2}=\frac{\csc \theta-\cot \theta}{2 \csc \theta} $$

Verify each identity. \(\frac{\sin x+\cos x}{\sin x}-\frac{\cos x-\sin x}{\cos x}=\sec x \csc x\)

verify each identity. $$ 4 \cos x \cos 2 x \sin 3 x=\sin 2 x+\sin 4 x+\sin 6 x $$

Find the exact value of the following under the given conditions: a. \(\cos (\alpha+\beta)\) b. \(\sin (\alpha+\beta)\) c. \(\tan (\alpha+\beta)\) \(\cos \alpha=\frac{8}{17}, \alpha\) lies in quadrant IV, and \(\sin \beta=-\frac{1}{2}, \beta\) lies in quadrant III.

Solve each equation on the interval \([0,2 \pi)\) $$ \tan ^{2} x \cos x=\tan ^{2} x $$

Verify each identity. $$ \cos ^{2} \frac{\theta}{2}=\frac{\sin \theta+\tan \theta}{2 \tan \theta} $$

Find the exact value of the following under the given conditions: a. \(\cos (\alpha+\beta)\) b. \(\sin (\alpha+\beta)\) c. \(\tan (\alpha+\beta)\) \(\cos \alpha=\frac{1}{2}, \alpha\) lies in quadrant IV, and \(\sin \beta=-\frac{1}{3}, \beta\) lies in quadrant III.

Solve each equation on the interval \([0,2 \pi)\) $$ \cot ^{2} x \sin x=\cot ^{2} x $$