Solving Trigonometric Equations $\cos{2x}=\cos{x}$

Question

How do you solve $\cos{2x}=\cos{x}$?

I'm really stuck, I know the first step is to isolate the variables, but after that I don't know what to do.

Answer 1

$$\implies2x =2n\pi\pm x$$ where $n$ is any integer

See also: this and $\sin y=0\implies y=m\pi$

Answer 2

$2\cos^2x-1=\cos x$. This is two degree equation. So

$$(2\cos x+1)(\cos x-1)=0$$

$$\cos x=1,-\dfrac{1}{2}$$

$$x=2nπ,\dfrac{2π}3+2nπ,\dfrac{4π}3+2nπ$$