Answer
$$\cos\frac{7\pi}{12}=\frac{\sqrt2-\sqrt6}{4}$$
Work Step by Step
$$\cos\frac{7\pi}{12}$$
In terms of radians, the angles $\frac{\pi}{6}$, $\frac{\pi}{4}$ and $\frac{\pi}{3}$ are already known. So we would try to rewrite $\frac{7\pi}{12}$ in terms of them.
$$\frac{7\pi}{12}=\frac{3\pi+4\pi}{12}=\frac{3\pi}{12}+\frac{4\pi}{12}=\frac{\pi}{4}+\frac{\pi}{3}$$
That means,
$$\cos\frac{7\pi}{12}=\cos(\frac{\pi}{4}+\frac{\pi}{3})$$
We apply cosine of a sum:
$$\cos\frac{7\pi}{12}=\cos\frac{\pi}{4}\cos\frac{\pi}{3}-\sin\frac{\pi}{4}\sin\frac{\pi}{3}$$
$$\cos\frac{7\pi}{12}=\frac{\sqrt2}{2}\frac{1}{2}-\frac{\sqrt2}{2}\frac{\sqrt3}{2}$$
$$\cos\frac{7\pi}{12}=\frac{\sqrt2}{4}-\frac{\sqrt6}{4}$$
$$\cos\frac{7\pi}{12}=\frac{\sqrt2-\sqrt6}{4}$$
