Answer
The required value is $\frac{1}{2}$
Work Step by Step
We have the difference formula, $\cos \alpha \cos \beta +\sin \alpha \sin \beta =\cos \left( \alpha -\beta \right)$. Now, applying the difference formula where $\alpha $ is $\frac{5\pi }{12}$ and $\beta $ is $\frac{\pi }{12}$, we get:
$\begin{align}
& \cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12}=\cos \left( \frac{5\pi }{12}-\frac{\pi }{12} \right) \\
& =\cos \frac{4\pi }{12} \\
& =\cos \frac{\pi }{3} \\
& =\frac{1}{2}
\end{align}$
Hence, the required value is $\frac{1}{2}$
