Solve for P (complex solution)
\left\{\begin{matrix}P=-i\ln(\frac{\sqrt{2}\left(\sqrt{2}\sin(Y)-\sqrt{-\cos(2Y)-1}\right)}{2})+2\pi n_{1}\text{, }n_{1}\in \mathrm{Z}\text{, }&\frac{\sqrt{2}\left(\sqrt{2}\sin(Y)-\sqrt{-\cos(2Y)-1}\right)}{2}\neq 0\\P=-i\ln(\frac{\sqrt{2}\left(\sqrt{2}\sin(Y)+\sqrt{-\cos(2Y)-1}\right)}{2})+2\pi n_{2}\text{, }n_{2}\in \mathrm{Z}\text{, }&\frac{\sqrt{2}\left(\sqrt{2}\sin(Y)+\sqrt{-\cos(2Y)-1}\right)}{2}\neq 0\end{matrix}\right.
Solve for Y (complex solution)
\left\{\begin{matrix}Y=-i\ln(\frac{2i\cos(P)-\sqrt{2}\sqrt{-\cos(2P)+1}}{2})+2\pi n_{1}\text{, }n_{1}\in \mathrm{Z}\text{, }&\frac{2i\cos(P)-\sqrt{2}\sqrt{-\cos(2P)+1}}{2}\neq 0\\Y=-i\ln(\frac{2i\cos(P)+\sqrt{2}\sqrt{-\cos(2P)+1}}{2})+2\pi n_{2}\text{, }n_{2}\in \mathrm{Z}\text{, }&\frac{2i\cos(P)+\sqrt{2}\sqrt{-\cos(2P)+1}}{2}\neq 0\end{matrix}\right.
Solve for P
P=-\arccos(\sin(Y))+2\pi n_{1}\text{, }n_{1}\in \mathrm{Z}
P=\arccos(\sin(Y))+2\pi n_{2}\text{, }n_{2}\in \mathrm{Z}
Solve for Y
Y=-\arcsin(\cos(P))+2\pi n_{1}+\pi \text{, }n_{1}\in \mathrm{Z}
Y=\arcsin(\cos(P))+2\pi n_{2}\text{, }n_{2}\in \mathrm{Z}
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