Answer
The statement
$$\cos\frac{2\pi}{3}=\cos\frac{11\pi}{12}\cos\frac{\pi}{4}+\sin\frac{11\pi}{12}\sin\frac{\pi}{4}$$
is true.
Work Step by Step
$$\cos\frac{2\pi}{3}=\cos\frac{11\pi}{12}\cos\frac{\pi}{4}+\sin\frac{11\pi}{12}\sin\frac{\pi}{4}$$
We now try subtracting $\frac{\pi}{4}$ from $\frac{11\pi}{12}$,
$$\frac{11\pi}{12}-\frac{\pi}{4}=\frac{11\pi}{12}-\frac{3\pi}{12}=\frac{8\pi}{12}=\frac{2\pi}{3}$$
Hence, we can rewrite $\cos\frac{2\pi}{3}$:
$$\cos\frac{2\pi}{3}=\cos\Big(\frac{11\pi}{12}-\frac{\pi}{4}\Big)$$
Now we use the cosine difference identity
$$\cos(A-B)=\cos A\cos B+\sin A\sin B$$ (be extremely careful about the sign in the middle)
to expand $\cos\Big(\frac{11\pi}{12}-\frac{\pi}{4}\Big)$:
$$\cos\frac{2\pi}{3}=\cos\frac{11\pi}{12}\cos\frac{\pi}{4}+\sin\frac{11\pi}{12}\sin\frac{\pi}{4}$$
This means that the statement
$$\cos\frac{2\pi}{3}=\cos\frac{11\pi}{12}\cos\frac{\pi}{4}+\sin\frac{11\pi}{12}\sin\frac{\pi}{4}$$
is true.
