I am able to derive $$\sin^{-1}x+\sin^{-1}y=\sin^{-1}\left(x \sqrt{1-y^2}+y\sqrt{1-x^2}\right)$$ as follows.
Let $$z=\sin^{-1}{x}+\sin^{-1}y.$$ Now if $z \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
taking $\sin$ on both sides we get
$$\sin z=x \sqrt{1-y^2}+y\sqrt{1-x^2}.$$
Now taking $\sin^{-1}$ on both sides we get
$$\sin^{-1}(\sin z)=\sin^{-1}\left(x \sqrt{1-y^2}+y\sqrt{1-x^2}\right).$$ But in $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ we have $\sin^{-1}(\sin z)=z.$ Hence
$$z=\sin^{-1}\left(x \sqrt{1-y^2}+y\sqrt{1-x^2}\right)$$ but my book gave this is valid for $x^2+y^2 \le 1$ or $x^2+y^2 \gt 1$ when $xy \gt 0$. Can I know why?
