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 6 of 6: State the odd-nn solutions
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n odd:x=2kπorx=3π2+2kπ,k∈Zn\text{ odd}:\quad x=2k\pi\quad\text{or}\quad x=\frac{3\pi}{2}+2k\pi,\qquad k\in\mathbb Z
Detailed analysis

The sign intervals cover one full period, and the equation is 2π2\pi-periodic. Therefore for odd nn the complete solution set is x=2kπx=2k\pi or x=3π/2+2kπx=3\pi/2+2k\pi.