Answer
The exact value of the provided expression is $\frac{\sqrt{2}}{2}$.
Work Step by Step
Let us assume $\theta ={{\sin }^{-1}}\left( \frac{\sqrt{2}}{2} \right)$. Then, we have
$\sin \theta =\left( \frac{\sqrt{2}}{2} \right)$
We know that for the sine function, the interval for the angle is $\left[ -\frac{\pi }{2},\frac{\pi }{2} \right]$.
So, the only angle that satisfies $\sin \theta =\left( \frac{\sqrt{2}}{2} \right)$ is $\frac{\pi }{4}$.
Hence, $\theta =\left( \frac{\pi }{4} \right)$ and the exact value of the given expression is
$\begin{align}
& \cos \left( {{\sin }^{-1}}\frac{\sqrt{2}}{2} \right)=\cos \left( \theta \right) \\
& =\cos \left( \frac{\pi }{4} \right) \\
& =\frac{\sqrt{2}}{2}
\end{align}$
