$( X, Y)$ have a bivariate normal density centered at the origin with $E(X^2)$ = $E(Y^2) = 1$, and $E(XY) = p$ . In polar coordinates $(X, Y)$ becomes $(R,\Phi)$ where $R^2 = X^2 + Y^2$. Prove that $\Phi$ has a density given by
$$\frac{\sqrt{1-p^2}}{2\pi(1-2p\sin(\varphi)\cos(\varphi))}$$ And is uniformly distributed iff $p = 0$. (To this point everything is clear)
what i do not understand is how to conclude that $P\{XY > 0\} = \frac{1}{2} +\pi^{-1} \arcsin (p)$ and $P\{XY < 0\}= \pi^{-1} \arccos (p)$.
