Let $U_1$ and $U_2$ be uniformly distributed on $(0,1]$.
Let $$ X_1 = \sqrt{-2\log(U_1)} \cos(2 \pi U_2), $$ and
$$ X_2 = \sqrt{-2\log(U_1)} \sin(2 \pi U_2) $$
Find the joint distribution of $X_1$ and $X_2$.
I can see that $X_1^2 + X_2^2 = -2log(U_1)$ and $\frac{X_2}{X_1} = \tan(2 \pi U_2) $ but I don't know how to further manipulate the transformation to get the joint distribution.