Question

Prove that : $\tan 8\theta -\tan 6\theta -\tan 2\theta =\tan 8\theta .\tan 6\theta \tan 2\theta$

Answer 1

$LHS=\tan 8\theta -\tan 6\theta -\tan 2\theta$

We know $\tan \left(8\theta -6\theta \right)=\frac{\tan 8\theta -\tan 6\theta }{1+\tan 8\theta +\tan 6\theta }$

$\Rightarrow \tan 8\theta -\tan 6\theta =\tan 2\theta \left(1+\tan 8\theta \tan 6\theta \right)$

$LHS=\tan 2\theta \left(1+\tan 8\theta \tan 6\theta \right)-\tan 2\theta$

$=\tan 2\theta +\tan 2\theta \tan 8\theta \tan 6\theta -\tan 2\theta$

$=\tan 8\theta .\tan 6\theta \tan 2\theta$