Answer
The exact value of $2{{\cos }^{2}}\frac{\pi }{8}-1$ is $\frac{\sqrt{2}}{2}$.
Work Step by Step
Recall the given expression.
$\cos 2\theta =2{{\cos }^{2}}\theta -1$
Apply the given expression.
$\begin{align}
& 2{{\cos }^{2}}\frac{\pi }{8}-1=\cos 2\left( \frac{\pi }{8} \right) \\
& =\cos \left( \frac{\pi }{4} \right) \\
& =\frac{1}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}} \\
& =\frac{\sqrt{2}}{2}
\end{align}$
Therefore, the exact value of $2{{\cos }^{2}}\frac{\pi }{8}-1$ is $\frac{\sqrt{2}}{2}$.
