Answer
$\frac{1}{2}$
Work Step by Step
We use the subtraction formula for cosine, $\cos(s-t)=\cos s\cos t+\sin s\sin t$, with $s=\frac{3\pi}{7}$ and $t=\frac{2\pi}{21}$.
$\cos(\frac{3\pi}{7}) \cos(\frac{2\pi}{21})+\sin(\frac{3\pi}{7})\sin(\frac{2\pi}{21})$
$=\cos(\frac{3\pi}{7}-\frac{2\pi}{21})$
$=\cos(\frac{9\pi}{21}-\frac{2\pi}{21})$
$=\cos(\frac{7\pi}{21})$
$=\cos(\frac{\pi}{3})$
$=\frac{1}{2}$
