Answer
The domain is
$$\mathcal{D}=\left\{(x,y)\left|-\frac{\pi}{2}\leq x+y\leq\frac{\pi}{2}\right.\right\}$$
and it is shown on the figure below
Work Step by Step
The argument of the $\sin^{-1}$ function has to be from the segment $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ so we need that $x+y\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$, This can be rewritten as
$$-\frac{\pi}{2}\leq x+y\leq\frac{\pi}{2}$$
and then as two inequalities:
$$x+y\geq-\frac{\pi}{2}\Rightarrow y\geq-x-\frac{\pi}{2};$$
and
$$x+y\leq\frac{\pi}{2}\Rightarrow y\leq-x+\frac{\pi}{2}.$$
Thus, the doman is the region bounded by two parallel lines (including them): $y=-x-\pi/2$ and $y=-x+\pi/2$.
So we write for the domain
$$\mathcal{D}=\left\{(x,y)\left|-\frac{\pi}{2}\leq x+y\leq\frac{\pi}{2}\right.\right\}$$
and it is presented in the figure below.