 Solution

The Box-Muller transform uses independent uniforms to setThe radius has the unit Rayleigh distribution, with density $r{e}^{-r^2/2}$ for $r>0$, and the angle is independently uniform on $[0,2\pi )$. Their joint density is $(2\pi {)}^{-1}r{e}^{-r^2/2}$. The Cartesian change of variables has absolute Jacobian determinant $r$, so the joint density of $(X_1,X_2)$ isThe factorization proves that the outputs are independent random variables, each with a standard normal distribution.