Find the general solution for tan x = tan 4x

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I am studying maths as a hobby and have come to the following problem:

Find the general solution for $\tan x = \tan 4x$

My book says the general form of such an answer is:

$\tan x = \tan \alpha \implies x = n\pi +\alpha$

When I look at the answer at the back of the book it says it is

$\frac{n\pi}{3}$, which is not in the same format.

In any case, I cannot see how this was worked out. I have tried using double angle formulae but it gets very messy and I get confused.

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1 Answer

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$$\tan (x) = \tan (4x)\xrightarrow{{n \in \mathbb{Z}}}4x = n\pi + x \to 3x = n\pi \to x = \frac{{n\pi }}{3}$$

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