Question

Cot x sec^4 x=cot x + 2tan x + tan^3 x

Answer 1

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.