\(\displaystyle f_{XY}(x,y) = f_{X}(x)f_{Y}(y) = \frac{1}{2\pi \sigma^2} e^{-\frac{x^2+y^2}{2\sigma^2}}\)

I have to show that \(\displaystyle Z\) is also a normal random variable

\(\displaystyle Z = \frac{(X-Y)^2-2Y^2}{\sqrt{X^2+Y^2}}\)

maybe using these substitions

\(\displaystyle A = \sqrt {X^2+Y^2}\)

\(\displaystyle \tan B = \frac{Y}{X}\)