Answer
The value of $\cos \frac{5\pi }{12}$ is $\frac{\sqrt{6}-\sqrt{2}}{4}$
Work Step by Step
By using the trigonometric identity, we get
$\cos \left( A+B \right)=\cos A\cos B-\sin A\sin B$
So,
$\begin{align}
& \cos \frac{5\pi }{12}=\cos \left( \frac{\pi }{6}+\frac{\pi }{4} \right) \\
& =\cos \frac{\pi }{6}\cdot \cos \frac{\pi }{4}-\sin \frac{\pi }{6}\cdot \sin \frac{\pi }{4} \\
& =\frac{\sqrt{3}}{2}\cdot \frac{1}{\sqrt{2}}-\frac{1}{2}\cdot \frac{1}{\sqrt{2}} \\
& =\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}
\end{align}$
Therefore,
$\cos \frac{5\pi }{12}=\frac{\sqrt{6}-\sqrt{2}}{4}$
