Solution of $\tan(nx)=k\tan(x)$ Could you kindly suggest me ways to find out closed form solution of the following equation type:
$$\tan(nx)=k\tan(x),$$
where $n$ and $k$ are some positive real numbers (that excludes zero). I can solve it using series expansion, but that gives me only an approximate solution for $x$.
Thanks in advance.
PS: I am in engineering stream, but I usually don't need to solve problems that seems very tough to me, mostly just couple of additions/subtractions etc.

If $n$ and $k$ are positive integers $ \tan n x $ can be expressed as a fraction as indicated by Claude Leibovici. A common factor $ \tan x $ can be cancelled out, you have an $ n^{th} $ degree polynomial:
$$ \dfrac{\tan(nx)} {\tan(x)} - k = 0 $$ 
Else numerical solutions by Newton iteration etc. with  correct seed value so that you do not land up in any of $ \pi$ intervals of argument.
