Question

Sin 2x + tan 2x = 4tan/ 1- tan^4 prove that

Answer 1

`(4\tan x)/(1-\tan^4x)=(4\tan x)/((1-\tan^2x)(1+\tan^2x))`

`=(4\tan x)/((1-(\sec^2x-1)\sec^2x)`

`=(4\tan x\cos^2x)/(2-\sec^2x)`

`=(4\sin x\cos x)/(2-\sec^2x)`

`=(4\sin x\cos^3x)/(2\cos^2x-1)`

`=(2\sin2x\cos^2x)/(\cos2x)`

`=2\tan2x\cos^2x`

`=2\tan2x\left(\frac{1+\cos2x}2\right)`

`=\tan2x(1+\cos2x)`

`=\tan2x+\sin2x`