Aus Online Mathematik Brückenkurs 1

Let's first investigate when the equality

\displaystyle \tan u=\tan v

is satisfied. Because \displaystyle \tan u can be interpreted as the slope (gradient) of the line which makes an angle u with the positive x-axis, we see that for a fixed value of \displaystyle \tan u, there are two angles v in the unit circle with this slope,

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

The angle v has the same slope after every half turn, so if we add multiples of \displaystyle \pi to u, we will obtain all the angles v which satisfy the equality

\displaystyle v=u+n\pi\,,

where n is an arbitrary integer.

If we apply this result to the equation

\displaystyle \tan x=\tan 4x

we see that the solutions are given by

\displaystyle 4x = x+n\pi\qquad\text{(n is an arbitrary integer),}

and solving for x gives

\displaystyle x = \tfrac{1}{3}n\pi\qquad\text{(n is an arbitrary integer).}