From Förberedande kurs i matematik 1

For a fixed value of u, an equality of the form

\displaystyle \cos u=\cos v

is satisfied by two angles v in the unit circle,

\displaystyle v=u\qquad\text{and}\qquad v=-u\,\textrm{.}

This means that all angles v which satisfy the equality are

\displaystyle v=u+2n\pi\qquad\text{and}\qquad v=-u+2n\pi\,,

where n is an arbitrary integer.

Therefore, the equation

\displaystyle \cos 5x=\cos (x+\pi/5)

has the solutions

\displaystyle \left\{\begin{align} 5x&=x+\frac{\pi}{5}+2n\pi\quad\text{or}\\[5pt] 5x &= -x-\frac{\pi}{5}+2n\pi\,\textrm{.}\end{align}\right.

If we collect x onto one side, we end up with

\displaystyle \left\{\begin{align} x &= \frac{\pi}{20} + \frac{n\pi}{2}\,,\\[5pt] x &= -\frac{\pi }{30}+\frac{n\pi}{3}\,, \end{align}\right.

where n is an arbitrary integer.