\(\dfrac{4\tan x}{1-\tan^4x}=\dfrac{4\tan x}{(1-\tan^2x)(1+\tan^2x)}\)
\(=\dfrac{4\tan x}{(1-(\sec^2x-1)\sec^2x}\)
\(=\dfrac{4\tan x\cos^2x}{2-\sec^2x}\)
\(=\dfrac{4\sin x\cos x}{2-\sec^2x}\)
\(=\dfrac{4\sin x\cos^3x}{2\cos^2x-1}\)
\(=\dfrac{2\sin2x\cos^2x}{\cos2x}\)
\(=2\tan2x\cos^2x\)
\(=2\tan2x\left(\dfrac{1+\cos2x}2\right)\)
\(=\tan2x(1+\cos2x)\)
\(=\tan2x+\sin2x\)
