Answer
$\frac{\sqrt{6}}{2}$
Work Step by Step
Use the third Sum-to-Product Formula, $\cos(x+y)=2\cos \frac{x+y}{2}\cos \frac{x-y}{2}$.
$\cos\frac{\pi}{12}+\cos \frac{5\pi}{12}$
$=2\cos\frac{\frac{\pi}{12}+\frac{5\pi}{12}}{2}\cos\frac{\frac{\pi}{12}-\frac{5\pi}{12}}{2}$
$=2\cos \frac{\frac{\pi}{2}}{2}\cos\frac{-\frac{\pi}{3}}{2}$
$=2\cos \frac{\pi}{4}\cos(-\frac{\pi}{6})$
$=2\times\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}$
$=\frac{\sqrt{6}}{2}$
