13. Solve the equation $\cos ^{n} x-\sin ^{n} x=1$, where $n$ is any given positive integer.
13. For $n \geqslant 2$, $\cos ^{n} x-\sin ^{n} x \leqslant \cos ^{2} x+\sin ^{2} x=1$, equality holds only when $\cos x= \pm 1$ or 0. Therefore, when $n$ is odd, $x=2 k \pi$ or $2 k \pi-\frac{\pi}{2}$; when $n$ is even, $x=k \pi$, where $k$ is any integer. For $n=1$, $\cos x-\sin x=1$, squaring and rearranging gives $\sin 2 x=0$. Thus, the solutions in this case are also included in the scenario where $n$ is odd.