Prove that: tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x

Question

Prove that:

\[ \tan 8x-\tan 6x-\tan 2x = \tan 8x\tan 6x\tan 2x \]

Proof

\[ \tan 8x = \tan(6x+2x) \]

\[ = \frac{\tan 6x+\tan 2x} {1-\tan 6x\tan 2x} \]

\[ \tan 8x(1-\tan 6x\tan 2x) = \tan 6x+\tan 2x \]

\[ \tan 8x-\tan 8x\tan 6x\tan 2x = \tan 6x+\tan 2x \]

\[ \tan 8x-\tan 6x-\tan 2x = \tan 8x\tan 6x\tan 2x \]

Hence proved.