Finding a pdf of a Gamma Distribution when $Z=\mathrm{arccot}\,(z)$

Question

I have been given $X \sim \Gamma(\alpha, \beta)$, so that $X$ has pdf $$f(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}\exp(−\beta x)$$ for $x > 0$ and $0$ otherwise. I have to find a pdf when $Z = \text{arccot}(x).$

I know for a fact that you have to do $f_Z(z)=f_X(\cot z)\cdot \lvert\cot'(z)\rvert$ but cant seem to get any further so any help will be appreciated.

Answer 1

There is nothing more to do, except for saying that $0<z<\frac{\pi}{2}$