Distribution of $Y_1$ and $Y_2$

$ X_1$ and $ X_2$ be two independent Normal $ (0,\sigma^2)$ random variables. Define $ Y_1={\sqrt{X_1^2 +X_2^2}}$ and $ Y_2=\frac{X_1}{\sqrt{Y_1}}$ . Show that $ Y_1$ and $ Y_2$ are independently distributed. I tried to use transformation technique $ (X_1,X_2) \to (Y_1,Y_2)$ but as the mapping is not bijective, I got confused. Please help me out.