Example 2. Find the extremum of $f(\alpha)=\cos \alpha+\cos \frac{\pi}{12}-\cos \left(\alpha+\frac{\pi}{12}\right)$.
Using an example similar to solving, the minimum value is $\cos \frac{\pi}{12}-$ $\sqrt{2-2 \cos \frac{\pi}{12}}$, the maximum value is $\cos \frac{\pi}{12}+\sqrt{2-2 \cos \frac{\pi}{12}}$.