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