#### Answer

$\frac{\sin^2 x}{2-2\cos x}=\cos^2\frac{x}{2}$

#### Work Step by Step

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.