if

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

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

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

$\displaystyle \tan B = \frac{Y}{X}$