## Precalculus (6th Edition)

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