IMO 1961 - Problem 3

Problem: Solve the equation $cos^{n}(x)-sin^{n}(x)=1$, where $n$ is a given positive integer.
Solution: For $n\ge 2$ we have:
$1=cos^{n}(x)-sin^{n}(x)\le |cos^{n}(x)-sin^{n}(x)|\le |cos^{n}(x)|+|sin^{n}(x)|\le cos^2(x)+sin^2(x)=1$.
Hence $sin^2(x)=|sin^{n}(x)|$ and $cos^2(x)=|cos^{n}(x)|$, from which it follows that $sin(x),cos(x)\in \left\{1,0,-1\right\}\implies x\in \frac{\pi\mathbb{Z}}{2}$. By inspection one obtains the set of solutions
 $\left\{m\pi |m\in \mathbb{Z}\right\}$ for even $n$ and $\left\{2m\pi,2m\pi-\frac{\pi}{2}|m\in \mathbb{Z}\right\}$ for odd $n$.
For $n=1$ we have $1=cos(x)-sin(x)=-\sqrt{2}sin(x-\frac{\pi}{4})$, which yields the set of solutions $\left\{2m\pi, 2m\pi- \frac{\pi}{2}|m\in \mathbb{Z}\right\}$.