## Precalculus (6th Edition)

$\tan 8\theta-\tan 8\theta\tan^2 4\theta=2\tan 4\theta$
Start with the left side: $\tan 8\theta-\tan 8\theta\tan^2 4\theta$ Factor: $=\tan 8\theta(1-\tan^2 4\theta)$ Rewrite $\tan 8\theta$ as $\tan(2*4\theta)$ and use the double-angle identity: $=\tan(2*4\theta)(1-\tan^2 4\theta)$ $=\frac{2\tan 4\theta}{1-\tan^2 4\theta}*(1-\tan^2 4\theta)$ Simplify: $=2\tan 4\theta$ Since this equals the right side, the identity has been proven.