#### Answer

$\tan \theta\sin 2\theta=2-2\cos^2 \theta$

#### Work Step by Step

Simplify the left side:
$\tan \theta\sin 2\theta$
$=\frac{\sin\theta}{\cos\theta}*2\sin\theta\cos\theta$
$=2\sin^2\theta$
Simplify the right side:
$2-2\cos^2 \theta$
$=2(1-\cos^2 \theta)$
$=2\sin^2\theta$
Since both the left side and the right side are equal to $2\sin^2\theta$, they are equal to each other, and the identity has been proven.