Recall the Pythagorean identity:
\(\sin^2x+\cos^2x=1\)
Divide through by \(\cos^2x\) and you get
\(\tan^2x+1=\sec^2x\)
On the left hand side of your equation, use this expansion for \(\sec^4x\):
\(\cot x\sec^4x=\cot x(\sec^2x)^2=\cot x(\tan^2x+1)^2=\cot x\tan^4x+2\cot x\tan^2x+\cot x\)
and simplify using the fact that \(\cot x=\dfrac1{\tan x}\). You end up with
\(\cot x\tan^4x+2\cot x\tan^2x+\cot x=\tan^3x+2\tan x+\cot x\)
so this is an identity and holds for all \(x\) in an appropriate domain.
