Answer
$$\dfrac{1}{2}$$
Work Step by Step
We know that $\cos$, $\sec$ are even trigonometric functions.
This implies that $f(-\theta)=f(\theta)$
So, $\cos(-\theta)=\cos(\theta)$
Apply the difference formula:
$\cos(A-B)=\cos(A)\cos(B)+\sin(A)\sin(B)$
Therefore,
$$\cos (\dfrac{\pi}{12}) \ \cos(\dfrac{5 \pi}{12}) +\sin (\dfrac{ \pi}{12}) \ \sin(\dfrac{5 \pi}{12})\\
=\cos \ (\dfrac{\pi}{12} -\dfrac{5 \pi}{12})\\
=\cos \ (\dfrac{-4 \pi}{12})\\
=\cos (\dfrac{\pi}{3}) \\=\dfrac{1}{2}$$
