Answer
$\dfrac{\sqrt2}{2}$
Work Step by Step
Note that $\sin^{-1}{\left(\frac{\sqrt2}{2}\right)}=\dfrac{\pi}{4}$, because $\sin{\left(\frac{\pi}{4}\right)}=\frac{\sqrt2}{2}$ and $\dfrac{\pi}{4}$ is in the range of $\sin^{-1}{x}$, which is $\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right].$
Thus,
$\cos{\left(\sin^{-1}{\left(\dfrac{\sqrt2}{2}\right)}\right)}=\cos{\dfrac{\pi}{4}}=\dfrac{\sqrt2}{2}$