Trigonometric Equations

Solve: $\tan 2x = \tan x$

$\tan 2x = \tan x$

i.e. $\;$ $\dfrac{2 \tan x}{1 - \tan^2 x} = \tan x$

i.e. $\;$ $2 \tan x = \tan x - \tan^3 x$

i.e. $\;$ $\tan^3 x + \tan x = 0$

i.e. $\;$ $\tan x \left(\tan^2 x + 1\right) = 0$

i.e. $\;$ $\tan x = 0$ $\;$ or $\;$ $1 + \tan^2 x = 0$

But $1 + \tan^2 x = 0$ is not possible $\because$ $\tan^2 x \neq -1$

Now, $\tan x = 0 \implies x = n \pi, \;\;\; n \in Z$