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=\frac1{\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.