## Precalculus (6th Edition)

$\frac{\sin^2 x}{2-2\cos x}=\cos^2\frac{x}{2}$
Simplify the left side: $\frac{\sin^2 x}{2-2\cos x}$ $=\frac{1-\cos^2 x}{2(1-\cos x)}$ $=\frac{(1+\cos x)(1-\cos x)}{2(1-\cos x)}$ $=\frac{1+\cos x}{2}$ Simplify the right side: $=\cos^2 \frac{x}{2}$ $=(\pm\sqrt{\frac{1+\cos x}{2}})^2$ $=\frac{1+\cos x}{2}$ Since the left side and the right side simplify to the same expression, the expressions are equal and the identity is proven.