What is cosX-cosY when X and Y are uniform random variables

Question

Let $X$ and $Y$ be iid uniform $$X,Y \sim U[-\pi,\pi]$$

Consider the following $$ U = cos (X)$$ $$V=cos(Y)$$ What is the distribution of $$W=U-V$$

I know that $$f_{U}(u) = \frac{1}{\pi\sqrt{1-u^2}}\hspace{1cm} -1 \leq u\leq 1$$ $$f_{V}(v) = \frac{1}{\pi\sqrt{1-v^2}}\hspace{1cm} -1 \leq v\leq 1$$

So then should I be taking the convolution $$\int_{-1}^{1} f_{U}(w+v) f_{V}(v)dv $$ Is this the right way to go? Any help is appreciated.

Thanks

Answer 1

You will probably have a better luck going to the Fourier transforms of the densities (use contour integration), multiplying them, then using the inverse Fourier transform to get the density of the difference (see Characteristic function).