Question

I need help proving this trigonometric equation:
cotx sec^(4)x = cotx + 2tanx + tan^(3)x
I started with the right side, but the farthest I have gotten is:
(1/tanx) + 2tanx + tan^(3)x
What next?

Answer 1

\[\cot(x)\sec^4(x)=\cot(x)(\sec^2(x))^2=\cot(x)(1+\tan^2(x))^2\]\[\cot(x)(1+2\tan^2(x)+\tan^4(x))=\cot(x)+2\tan(x)+\tan^3(x)\]

Answer 2

I also did it by setting up both sides in terms of sin and cos, which works but is not nearly as elegant.