It is well known that
$$\sum_{n=0}^{\infty} a^n \cos(n\theta) = \frac{1-a\cos(\theta)}{1-2a\cos(\theta)+a^2}$$ $$\sum_{n=0}^{\infty} a^n \sin(n\theta) = \frac{a\sin(\theta)}{1-2a\cos(\theta)+a^2}$$
which come naturally from expanding $\sum a^n e^{in\theta}$ and comparing real and imaginary components. Inspired by this, I decided to pursue the following as a first step to deriving similar closed forms (hopefully) for other trigonometric functions:
$$f(\theta,a) = \sum_{n=0}^{\infty} a^n \tan(n\theta)$$
Admittedly, this isn't as tractable as the above two; through sum manipulation, using Ramanujan's master theorem, and inversion Mellin transform, I derived the following
$$f(\theta,a) = \lim_{T \to \infty}\frac{1}{2\pi i} \int_{c-iT}^{c+iT} \theta^{-s} \frac{(1-2^{1-s})\Gamma(s)\zeta(s)\mathrm{Li}_{s}(a)}{(2i)^{s-1}} \ ds$$
where $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$ and $$\mathrm{Li}_{s}(a) = \sum_{n=1}^\infty \frac{a^n}{n^s}$$ are the Riemann zeta function and Polylogarithm function, respectively. I am sure there is literature somewhere observing these kind of problems, yet I am having a hard time finding some. Anyone have a direction to point to, or pointing to closed forms? Much appreciated!