Problem 3

Solve the equation

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

, where

n

is a given positive integer.

Step 4 of 6: Exclude the third quadrant

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

Detailed analysis

Here both sine and cosine are negative. Put $u=|\sin x|$ and $v=|\cos x|$ , so $u,v\in(0,1)$ and $u^2+v^2=1$ . For odd $n$ , the expression is $u^n-v^n<u^n<1$ , so no point in this open interval solves the equation. The endpoint $x=3\pi/2$ gives equality.