What is the domain of the function $f(x)=\cos ({\sin }^{-1}(x+1))$?

The function $f(x)$ is defined when the argument of ${\sin }^{-1}$ is within the valid domain of the arcsine function.The domain of ${\sin }^{-1}(u)$ is $-1\le u\le 1$.Here $u=x+1$:
$$-1\le x+1\le 1$$
Subtract $1$ from all parts:
$$-1-1\le x\le 1-1$$
$$-2\le x\le 0$$
In interval notation, the domain is $[-2,0]$.