Differentiate 1/tanx

Question

I got -sec^2 / tan^2x, which is correct, but the answers said I could also simplify this to -cosec^2. How would you do that?

Thanks =)

Answer 1

-sec^2/tan^2 = -sec^2 * cos^2/sin^2 = -1/sin^2 = -csc^2

Answer 2

also, just as
tan' = sec^2
cot' = -csc^2 as you proved