Answer
$$\cos\frac{11\pi}{12}=\frac{-\sqrt2-\sqrt6}{4}$$
Work Step by Step
As suggested by the exercise, we would rewrite:
$$\frac{11\pi}{12}=\frac{\pi}{4}+\frac{2\pi}{3}$$
Therefore, $$\cos\frac{11\pi}{12}=\cos\Big(\frac{\pi}{4}+\frac{2\pi}{3}\Big)$$
Apply the addition formula for cosine here:
$$\cos\frac{11\pi}{12}=\cos\frac{\pi}{4}\cos\frac{2\pi}{3}-\sin\frac{\pi}{4}\sin\frac{2\pi}{3}$$
Remember that $\sin\frac{\pi}{4}=\cos\frac{\pi}{4}=\frac{\sqrt2}{2}$ and $\sin\frac{2\pi}{3}=\frac{\sqrt3}{2}$ and $\cos\frac{2\pi}{3}=-\frac{1}{2}$
So,
$$\cos\frac{11\pi}{12}=\frac{\sqrt2}{2}\times\Big(-\frac{1}{2}\Big)-\frac{\sqrt2}{2}\times\frac{\sqrt3}{2}$$
$$\cos\frac{11\pi}{12}=-\frac{\sqrt2}{4}-\frac{\sqrt6}{4}$$
$$\cos\frac{11\pi}{12}=\frac{-\sqrt2-\sqrt6}{4}$$
