Prove that: (tan²2x − tan²x)/(1 − tan²2x tan²x) = tan 3x tan x

Question

Prove that:

\[ \frac{\tan^22x-\tan^2x} {1-\tan^22x\tan^2x} = \tan3x\tan x \]

Proof

L.H.S.

\[ = \frac{(\tan2x-\tan x)(\tan2x+\tan x)} {(1-\tan2x\tan x)(1+\tan2x\tan x)} \]

\[ = \frac{\tan2x-\tan x} {1+\tan2x\tan x} \times \frac{\tan2x+\tan x} {1-\tan2x\tan x} \]

Using

\[ \tan(A-B) = \frac{\tan A-\tan B} {1+\tan A\tan B} \]

and

\[ \tan(A+B) = \frac{\tan A+\tan B} {1-\tan A\tan B} \]

\[ = \tan(2x-x)\tan(2x+x) \]

\[ = \tan x\tan3x \]

\[ = \tan3x\tan x \]

Hence proved.