Answer
$\dfrac{1}{2}$
Work Step by Step
Recall that:
$\cos(\alpha-\beta)=\cos(\alpha)\cos(\beta)+\sin(\alpha)\sin(\beta)$
Hence,
$\cos(\frac{\pi}{12})\cos(\frac{5\pi}{12})+\sin(\frac{\pi}{12})\sin(\frac{5\pi}{12})=\cos\left(\frac{\pi}{12}-\frac{5\pi}{12}\right)=\cos(-\frac{\pi}{3})=\frac{1}{2}$
