# Is transfert theorem the best choice in this kind of exercise?

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 ?

I didn't find a translation for that, it is called "Théorème du transfert" in french. $\mathbb E\left[\varphi(X)\right] \stackrel{\text{déf.}}{=} \int_\Omega \varphi \big(X(\omega)\big) \mathbb{P}(\mathrm d\omega) = \int_\mathbb{R} \varphi(x) \mathbb P_X(\mathrm dx)$ – Alan Simonin Jan 21 '13 at 0:55