Generally $ \tan(\alpha+\beta+\gamma+\cdots) = \frac{e_1-e_3+e_5-e_7+\cdots}{e_0 -e_2+e_4-e_6+\cdots}\tag{1} $ where $e_k$ is the sum of all products of $k$ of the tangents $\tan\alpha,\tan\beta,\tan\gamma,\ldots\ {}$. For example, if there are just four variables, $\alpha,\beta,\gamma,\delta$, then $ e_2 = \tan\alpha\tan\beta + \tan\alpha\tan\gamma + \tan\alpha\tan\delta + \tan\beta\tan\gamma+\tan\beta\tan\delta +\tan\gamma\tan\delta $ and $ e_3 = \tan\alpha\tan\beta\tan\gamma+\tan\alpha\tan\beta\tan\delta+\tan\alpha\tan\gamma\tan\delta+\tan\beta\tan\gamma\tan\delta, $ and so on. And of course $e_0=1$ (except when there are $0$ variables, in which case $e_0=0$). It's easy to prove $(1)$ by mathematical induction on the number of variables.
So if you want $\tan(n\alpha)$, it's just the case where all of the variables are the same variable, $\alpha$. So for example, if there are four variables, then $e_2 = \dbinom 4 2 \tan\alpha\tan\alpha = 6\tan^2\alpha$ and $e_3 = \dbinom 4 3 \tan^3\alpha$, etc.