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