 Radial-ratio independence of two half-normal variables

If $X,Y$ are independent with density $2e^{-x^2}/\sqrt{\pi }$ on the positive axis, the Jacobian matrix of the inverse transformation has determinant $1/[2(1+v^2)]$. The joint density is $2e^{-u}/[\pi (1+v^2)]$ for $u,v>0$, factoring into an exponential distribution for $U$ and a unit-scale half-Cauchy distribution for $V$. The underlying radial symmetry makes radius and angle independent.