<s>[INST] Solve the equation $\cos^n(x) - \sin^n(x)=1$ 
Solve the equation $\cos^n(x) - \sin^n(x)=1,n \in \mathbb{N}-\{0\}$


If $n$ is even then $\cos^n(x) = \sin^n(x)+1$ is only possible if $\sin(x)=0$ therefore the solution is $x=k\pi, k \in \mathbb{Z}$.
I'm having problems with $n$ odd case.
UPDATE
For $n=1$ we have $\cos(x) - \sin(x)=1$ and by squaring we get $sin(x)cos(x)=0$ which leads to $x=k\pi$ or $x=\pm \frac \pi 2 + k\pi$. From these solutions, only $x=2k\pi$ and $x=- \frac \pi 2 + 2k\pi$ are valid.
Also $x=2k\pi$ and $x=- \frac \pi 2 + 2k\pi$ are solutions for all odd $n$
 [/INST] I found a simple proof. For $n \ge 3$ odd we have $\cos^n(x) + (-\sin(x))^n=1 \tag1$ First, we know $\sin^2(x) + \cos^2(x)=1$ 
Also $|\cos^n(x)|\lt \cos^2(x)$ and $|\sin^n(x)|\lt \sin^2(x)$ for $x \not = k\pi$ and $x \not = \pm \frac \pi 2 + k\pi$ therefore the equality (1) cannot hold for $x \not = k\pi$ and $x  \not = \pm \frac \pi 2 + k\pi$ 
It's easy to conclude from here.
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