$\sin^n a + \cos^n a = 1$ is only true when $n=2$

Question

Prove that

$$\forall a\in\mathbb R:\quad\sin^n a + \cos^n a = 1$$ is only true when $n=2$

Answer 1

Simply because, if $a\not\equiv0 \mod\dfrac\pi 2 $, we have $\,\,0< \lvert\sin a\rvert<1$ and $\,\,0< \lvert\cos a\rvert<1$, so that: $$\sin^n a\le \lvert\sin a\rvert^n<\sin^2 a\quad\text{and}\quad\cos^n a\le \lvert\cos a\rvert^n<\cos^2 a $$ for all $n>2$.

Answer 2

Hint:

Set $a=\pi/4$ and then see where you end up with...

Answer 3

Rewrite equation in the form:

$$\cos^n(a) + sin^n(a) = cos^2(a) + sin^2(a)$$ $$\cos^2(a)(cos^{n-2}(a) - 1) + sin^2(a)(sin^{n-2}(a) - 1) = 0 $$

Left part is negative for $$n \not= 2$$