Expression for $\cos^{-1}x\pm\cos^{-1}y$ As mentioned in Proof for the formula of sum of arcsine functions $\arcsin x+\arcsin y$ for $\sin^{-1}x+\sin^{-1}y$
$$
\sin^{-1}x+\sin^{-1}y=
\begin{cases}
\sin^{-1}( x\sqrt{1-y^2} + y\sqrt{1-x^2}) \;\;;x^2+y^2 \le 1 \\\
\pi - \sin^{-1}( x\sqrt{1-y^2} + y\sqrt{1-x^2}) \;\;;x^2+y^2 > 1, 0< x,y \le 1\\
-\pi - \sin^{-1}( x\sqrt{1-y^2} + y\sqrt{1-x^2}) \;\;;x^2+y^2 > 1, -1< x,y \le 0
\end{cases}
$$
Can we have similar expression for $\cos^{-1}x\pm\cos^{-1}y$ ?

Alternatively, to get rid of all those cases, we can write:
$$\sin^{-1}x\pm\sin^{-1}y=\text{atan2}(x\sqrt{1-y^2}\pm y\sqrt{1-x^2},\ \sqrt{1-x^2}\sqrt{1-y^2}\mp xy)$$
and:
$$\cos^{-1}x\pm\cos^{-1}y=\text{atan2}(y\sqrt{1-x^2}\pm x\sqrt{1-y^2},\ xy\mp\sqrt{1-x^2}\sqrt{1-y^2})$$
See atan2.
