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I know there are standard techniques to expand $\sin^n \theta$ and $\cos^n \theta$ $(n \in \mathbb{N})$ in terms of sines and cosines of multiples of $\theta$.

We take a complex number $z = \cos \theta + i \sin \theta$. Then

$$ \cos k\theta = {1 \over 2} \left ( z^k + {1 \over z^k} \right ) $$

where $k \in \mathbb{N}$ and

$$ \cos^n \theta = {1 \over 2^n} \left ( z + {1 \over z} \right )^n . $$

You just expand the second equation and repeatedly apply the first, to get your desired result. We do a similar thing for $\sin^n \theta$.

My question is: Is there any standard technique to expand $\tan^n \theta$ as a sum of tangents or sines or cosines of multiples of $\theta$?

EDIT: I noticed that an expansion like what I want doesn't exist even for $\tan^2 \theta$. The best we can do is: $\tan^2 \theta = 1 - 2 {\tan \theta \over \tan 2\theta}$. Now that it seems like an expansion might not exist, can anyone please justify why it doesn't exist? But if something does exist, then what is it? I'd prefer the expansion to be something which is easily integrable, just like with the expansions for $\sin^n \theta$ and $\cos^n \theta$.

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You can use this way to get $\sin^n\theta$ and $\cos^n\theta$ in terms of $\sin\theta$ and $\cos\theta,$ then $\tan^n\theta=\frac{\sin^n\theta}{\cos^n\theta}$ can be obtained from above, if your goal is just to expand it as a sum of tangents or sines or cosines of multiples of $\theta.$ (this result involves $\sin$ and $\cos$ terms.

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    I mean, I don't want an ugly fraction with sums of sines and cosines in the numerator and denominator. – 2017-01-01