MathLabs

Problem 3

Solve the equation cos⁡nx−sin⁡nx=1\cos^n x-\sin^n x=1, where nn is a given positive integer.
Step 1 of 6: Handle even nn
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In plain words

Even powers are nonnegative, so subtracting the sine term can only decrease the cosine term.

n even:cos⁡nx−sin⁡nx≤cos⁡nx≤1n\text{ even}:\quad \cos^n x-\sin^n x\le\cos^n x\le1
Detailed analysis

For even nn, sin⁡nx≥0\sin^n x\ge0 and ∣cos⁡x∣≤1|\cos x|\le1, hence cos⁡nx−sin⁡nx≤1\cos^n x-\sin^n x\le1. Equality requires both sin⁡nx=0\sin^n x=0 and cos⁡nx=1\cos^n x=1, so sin⁡x=0\sin x=0 and ∣cos⁡x∣=1|\cos x|=1.