Problem 3

Solve the equation

\cos^n x-\sin^n x=1

, where

n

is a given positive integer.

Step 5 of 6: Exclude the fourth quadrant interior

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x\in(3\pi/2,2\pi):\quad \cos^n x-\sin^n x=\cos^n x+|\sin x|^n<1

Detailed analysis

In this interval put $u=\cos x$ and $v=|\sin x|$ ; both lie in $(0,1)$ and $u^2+v^2=1$ . For $n=1$ , $u+v>1$ in the interior. For odd $n\ge3$ , $u^n+v^n<u^2+v^2=1$ . Thus the only endpoint solution here is $x=3\pi/2$ (while $x=2\pi$ is the same as $0$ modulo $2\pi$ ).