KöMaL 2012 A560

Given a right circular cone with vertex $O$, and a fixed point $P$ in the base plane of the cone. Draw a line $x$ through $P$ which intersects the base circle of the cone at two points, $X_1$ and $X_2$. Show that $\tan\frac{\angle POX_1}2 \cdot \tan\frac{\angle POX_2}2$ does not depend on the choice of the line $x$.