Question

How do you solve 5(cos^3)x=5cos x over the interval (0,2pi)?

Answer 1

Any multiple of `\pi/2` in the interval solves the equation.

Explanation:

First note that the factor `5` can be divided out leaving

`\cos^3(x)=\cos(x)`

`\cos(x)-\cos^3(x)=0`

Factoring and using the Pythagorean identity `\sin^2(x)+\cos^2(x)=1`:

`\cos(x)(1-\cos^2(x))=\cos(x)\sin^2(x)=0`

So either `\sin(x)` or `\cos(x)` could be `0` making `x` a multiple of `\pi/2`.