Finding the pdf of $(X+Y)^2/(X^2+Y^2)$ where $X$ and $Y$ are independent and normal 
$X$ and $Y$ are iid standard normal random variables.  What is the pdf of $(X+Y)^2/(X^2+Y^2)$?

I am guessing you would transform into polar coordinates and go from there, but I am getting lost.  Do we need two variable transformations here? 

Hint. Write $(X, Y) = (R\cos\Theta, R\sin\Theta)$. Can you conclude that $\Theta$ is uniform on $[0,2\pi]$? And if this true, how $(X+Y)^2/(X^2+Y^2)$ is written in terms of $\Theta$?
