Question

Solve the equation cos^n x − sin^n x = 1, where n is a natural number.

Answer 1

For 0<x<2π and positive even n, the only solution for cosnx−sinnx=1 is π.

The argument is simple as 0≤cos^nx,sin^nx≤1 and hence cos^nx−sin^nx=1 iff cos^nx=1 and sin^nx=0.

Answer 2

Go with trigonometric identity..... Sin^2x + Cos^2x = 1.