Answer
The exact value of $1-2{{\sin }^{2}}\frac{\pi }{12}$ is $\frac{\sqrt{3}}{2}$.
Work Step by Step
Recall the given expression.
$\cos 2\theta =1-2{{\sin }^{2}}\theta $
Apply the given expression.
$\begin{align}
& 1-2{{\sin }^{2}}\frac{\pi }{12}=\cos 2\left( \frac{\pi }{12} \right) \\
& =\cos \left( \frac{\pi }{6} \right) \\
& =\frac{\sqrt{3}}{2}
\end{align}$
Therefore, the exact value of $1-2{{\sin }^{2}}\frac{\pi }{12}$ is $\frac{\sqrt{3}}{2}$.
