Example 6 Determine an integer $k$ not less than 1, such that the expression $\sin k x \cdot \sin ^{k} x+$ $\cos k x \cdot \cos ^{k} x-\cos ^{k} 2 x$ is independent of $x$.
Solution: When $x=0$, $f(x)=0$, i.e., $f(0)=0$.
Take $x=\frac{\pi}{k}$, then $\cos ^{k} \frac{\pi}{k}=\cos ^{k} \frac{2 \pi}{k}$.
Therefore, $k$ is odd, and $\cos \frac{2 \pi}{k}=\cos \left(\pi-\frac{\pi}{k}\right)$, thus we derive $\pi-\frac{\pi}{k}=2 n \pi \pm \frac{\pi}{k}$, i.e., $\frac{1}{k}=2 n-1$ or $\frac{3}{k}=1-2 n(n \in \mathbb{Z})$, solving for $k=1$ or 3.
When $k=1$, the original expression is not a constant, hence $k=3$.