This is one of the problem in the Allan Gut's Second Course for Probability.
Let $X_1$, $X_2$ be independent standard gaussian e.g. N(0,1).
Let $Y_1 = \frac{X_1^2 - X_2^2}{\sqrt{X_1^2 + X_2^2}}$, $Y_2 = \frac{2X_1X_2}{\sqrt{X_1^2 + X_2^2}}$. Show $Y_1$, $Y_2$ are independent standard normal.
It is easy to show $Y_1$, $Y_2$ has mean 0, variance 1 and covariance 0. However, I am having trouble show they are joint normal, since the transformation is non-linear.
We can write $(X_1,X_2)$ using polar coordinates as $(R\cos \theta,\,R\sin\theta).$ It can be shown that if $(X_1,\,X_2)$ is a standard bivariate normal, $R$ and $\theta$ are independent, $R^2\sim\chi^2_2$ and $\theta\sim\text{Unif}(0,2\pi).$
A reference can be found here.
Now use the given transformation. You will see that $Y_1=\dfrac{R^2(\cos^2\theta-\sin^2\theta)}{R}$ and $Y_2=\dfrac{2R^2\cos\theta\sin\theta}{R},$ that is, $Y_1=R\cos(2\theta)$ and $Y_2=R\sin(2\theta).$ Since $\theta\sim\text{Unif}(0,2\pi)$ the distribution of $(\cos(2\theta),\,\sin(2\theta))$ is same as that of $(\cos(\theta),\,\sin(\theta)).$ Thus $$(Y_1,Y_2)=(R\cos(2\theta),\,R\sin(2\theta))\stackrel{d}{=}(R\cos\theta,\,R\sin\theta)=(X_1,X_2).$$
The $\stackrel{d}{=}$ symbol means equality in distribution.