If $X, Y \sim N(0,1)$ are iid rv, then $U=\frac{X Y}{\sqrt{X^{2}+Y^{2}}} \sim N(0,1 / 4)$
Given that $X,Y$ are independent $N(0,1)$ , show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}}$ are independent $N(0,\frac{1}{4})$