I am studying Probability theory and came to this exercise :
Let $U,V$ be independent uniform random variables over $[0,1]$. Show that $X:=\cos(2\pi V)\sqrt{-2\ln U}$ and $Y:=\sin(2\pi V)\sqrt{-2\ln U}$ are independent random variable which are normal distribution $N(0,1)$.
The proof is given in the textbook and uses the transfert theorem. And therefore, the probability density function of $X,Y$ are deducted from the Jacobian of the transformation.
My question is : Is this the usual method to solve such problems ? Could we use instead the cumulative distribution function or the characteristic function ? Or any other trick ?
Thanks for your help