Let's try a different approach
$The domain of $y=sin^{-1}(x^2)$ is equal to the range of its inverse function, so let's find the inverse$ \\ x=sin^{-1}(y^2)\\ sinx=y^2\\y=\sqrt{sinx} \\ $We know the values of $sinx$ range from $ -1\le x \le1,$ however as it is in a square root y must be $\ge0.$...