$Z_1:=\sqrt{-2\log X} \cos(2\pi Y), Z_2:=\sqrt{-2\log X} \sin(2\pi Y)$ independent and normal I am looking for a nice proof of the following statement:
If $X,Y\sim U(0,1)$ are two independent uniformly distributed random variables, then
$$Z_1:=\sqrt{-2\log X} \cos(2\pi Y), \quad Z_2:=\sqrt{-2\log X} \sin(2\pi Y)$$
are two independent standard normal random variables.
I found (sort of) a proof here: http://www.math.nyu.edu/faculty/goodman/teaching/MonteCarlo2005/notes/GaussianSampling.pdf but I am not really familiar with polar coordinates. Is there any way around it?

We want $X,Y$ in terms of $Z_1,Z_2$ so that we can use the change of variables transformation and show that the joint density of $Z_1,Z_2$ can factor into separate densities of $Z_1$ and $Z_2$ to show they're independent.
Note that $\frac{Z_2}{Z_1}=\tan(2\pi Y)$ and since we know $\cos^2(x)+\sin^2(x)=1$, then we get $\frac{Z_1^2}{\log X}+\frac{Z_2^2}{\log X}=-2$ so $Z_1^2+Z_2^2=-2\log X$.
Then we solve for $X,Y$ to get $X=\exp\left( -\frac{1}{2}(Z_1^2+Z_2^2)\right)$ and $Y=\frac{1}{2\pi}\arctan\left(\frac{Z_2}{Z_1}\right)$.
By transformation of variables,
$f_{Z_1,Z_2}(z_1,z_2)\frac{1}{\sqrt{2\pi}}\exp(-\frac{1}{2}z_1^2)\frac{1}{\sqrt{2\pi}}\exp(-\frac{1}{2}z_2^2)$
