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.