If X and Y are iid N(0,1), then try showing is Rayleigh distributed and the angle is uniformly distributed. Also note that this means, given any rayleigh, one can construct two i.i.d normally distributed random variables such that their length has the distribution of the given rayleigh, and the angle is unif. distributed.Call this a "decomposition result".

Now it is straight forward (Use Jacobian!) to show that is Rayleigh distributed. Now observe that where R is rayleigh and the angle is uniformly distributed. Now by decomposition result, we are done.