Let $\alpha\sim U(0,2\pi),R\sim \textrm{Exp}(0.5)$ be independent. Prove $X=\sqrt R \cos\alpha$ and $Y=\sqrt R \sin \alpha$ are independent and have standard normal distribution $N(0,1)$.
It is easy to find the distributions and densities of $\sqrt R,\sin\alpha,\cos \alpha$:$\sqrt R$,$\sin\alpha$. But I have difficulty finding the densities of $X$ and $Y$, as I come across an integral I don't know how to solve.
How to proceed?
