#### Answer

$\tan 8\theta-\tan 8\theta\tan^2 4\theta=2\tan 4\theta$

#### Work Step by Step

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.