The random variables $X_1$ and $X_2$ are independent and $\mathcal N(0, 1)$-distributed. Set $$Y_1=\frac{X_1^2-X_2^2}{\sqrt{X_1^2+X_2^2}} $$ and $$Y_2=\frac{2X_1\cdot X_2}{\sqrt{X_1^2+X_2^2}} $$
I want to show that $Y_1$ and $Y_2$ are independent $\mathcal N(0, 1)$-distributed random variables.
My idea was to define $X_1$ and $X_2$ as functions of $Y_1$ and $Y_2$, find the Jacobian and use the fact that we know the joint distribution of $(X_1,X_2)$ to find $f_{Y_1,Y_2}$. If the density $f_{Y_1,Y_2}$ turns out to be multivariate normal with a covariance matrix being diagonal (uncorrelated), then we know that $Y_1$ and $Y_2$ are independent and normally distributed. Am I correct?
If yes, then I need help finding the Jacobian. I have a hard time defining $X_1$ and $X_2$ as functions of $Y_1$ and $Y_2$ and dealing with the fact that the transformation has to be bijective.
And of course, any other approach would be appreciated as well. Thanks