I have this random variable X that has a uniform distribution on [-pi, pi]. I want to find the pdf for Y = cosX.

Here is my work so far:

F_{y}(y)= P( Y < y ) = P ( cos X < y) =*P ( X < cos^{-1}(y) ) + **P ( X > cos^{-1}(y) ) (note: I got * since cos(x) is increasing on [-pi , 0] and ** since cos(x) is decreasing on [0 , pi]).

$\displaystyle =\displaystyle{\int_{-pi}^{cos^{-1}(y) } {\left( {1/2pi} \right)} dx + \int_{cos^{-1}(y)}^{pi} {\left( {1/2pi} \right)} dx}$

I am sure that I can't write this as one integral as the two cos^{-1}(y) are not the same (and if I did it looks like I'll get 1)

Since cos(x) is even I know that somehow these two integrals are equal but I just can't show it.

Can someone please give me a hint?