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Question

how to solve: $\tan 2x=\tan x$

At first by speculation I realized that if $x$ was $2$ and $0$ it would satisfy the question, but then I heard that the solution was $k\pi$ when $k$ is an integer.

How do I figure out that the solution to the problem is $k\pi$?

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    Do u know $\tan y=\tan A, y=n\pi+A$ – 2017-02-10
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    Do you know how to express $\tan 2x$ in terms of $\tan x$, or the double-angle formulae for $\sin$ and $\cos$)? – 2017-02-10

2 Answers 2

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Tangent is periodic with period $\pi$, so we have $$\tan(a)=\tan(b) \iff b=n\pi+a$$ In this case we find that $2x = n\pi + x \implies x=n\pi$

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    If $f$ is periodic of period $T$, then $f(a)=f(b)$ does not imply $b=a+n\,T$. – 2017-02-10
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Hint. What can you say about two angles $A$ and $B$ if $\tan A = \tan B$? Looking at the graph of the tangent function may help you answer this question.