Answer
$$\tan8\theta-\tan8\theta\tan^24\theta=2\tan4\theta$$
As proved below, the equation is an identity.
Work Step by Step
$$\tan8\theta-\tan8\theta\tan^24\theta=2\tan4\theta$$
The left side would be examined first to see if it is equal to the right side.
$$X=\tan8\theta-\tan8\theta\tan^24\theta$$
$$X=\tan8\theta(1-\tan^24\theta)$$
$$X=\tan(2\times4\theta)(1-\tan^24\theta)$$
As we rewrite $\tan8\theta$ into $\tan(2\times4\theta)$, we can apply the Double-Angle Identity for $\tan2x$, which states
$$\tan2x=\frac{2\tan x}{1-\tan^2x}$$
with $x=4\theta$
Therefore, $$\tan(2\times4\theta)=\frac{2\tan4\theta}{1-\tan^24\theta}$$
And $X$ would be
$$X=\frac{2\tan4\theta}{1-\tan^24\theta}(1-\tan^24\theta)$$
$$X=2\tan4\theta$$
So, $$\tan8\theta-\tan8\theta\tan^24\theta=2\tan4\theta$$
The left side is equal to the right side. The equation is surely an identity.
