What summations of elementary trig functions are known to have (elementary) closed forms?

Question

I've been trying to find a closed form of $\displaystyle \sum_k{\tan{(k)}}$ that contains only elementary functions, and I think I may be onto something. But rather than reinvent the wheel, I want to ensure that this isn't already known.

So, I am specifically interested in the sum of the tangent. However, if it's not too "overbroad", I'm wondering: Which trigonometric summations are known to have these closed forms of elementary functions?

My broader question is to ask about all basic trig functions: sine, arcsine, hyperbolic sine, and inverse hyperbolic sine and their related counterparts like cosine, tangent, secant, cosecant, and cotangent. Which, out of all of these functions, have known elementary closed forms?

If these are known, where I can find references on these sums?

As asked by J.E. Pascoe, I am interested in solutions of

$$\displaystyle \sum_{k=a}^{b}{ \tan{(k)} }$$

where $a$ and $b$ are taken to be naturals. However, if someone can find an answer in some wider domain, this is interesting to me as well.

Answer 1

From this answer of mine The antidifference of tangent is:

$$\sum_z\tan z=\sum _{k=1}^{\infty } \left(\psi \left(k \pi -\frac{\pi }{2}+1\right)+\psi \left(k \pi -\frac{\pi }{2}\right)-\psi \left(k \pi -\frac{\pi }{2}+1-z\right)-\psi \left(k \pi -\frac{\pi }{2}+z\right)\right)+C$$

A closed form for it symbolically resolves to

$$\sum_z \tan z=iz-\psi _{e^{2 i}}^{(0)}\left(z+\frac{\pi }{2}\right)+C$$

where the function involved is the q-digamma function (which is not elementary) although the usual definition of q-digamma does not converge at this base.

Thus the answer to your question is no, the antidifference of tangent is not elementary.

But you may be interested in talking about the antidifference taken with step $\pi$ which is more natural than step 1 when speaking about trigonometric functions.

In this case,

$$\Delta_\pi^{-1} \tan x = \frac x\pi \tan x + C$$

which is elementary. Similarly, the antidifference with step 1 of $\tan \pi x$ is also elementary:

$$\sum_x\tan \pi x=x \tan \pi x+C$$

The both identities follow from the period rule.

Regarding other elementary functions you can find some tables in Wikipedia.

Answer 2

Here are some formulas from Jolley, Summation of Series:

$$\sum_1^n\sin k\theta=\sin{(n+1)\theta\over2}\sin{n\theta\over2}\csc{\theta\over2}\tag{417}$$

$$\sum_1^n\cos k\theta=\cos{(n+1)\theta\over2}\sin{n\theta\over2}\csc{\theta\over2}\tag{418}$$

$$\sum_1^na^{k-1}\cosh(k-1)\theta={1-a\cosh\theta-a^n\cosh n\theta+a^{n+1}\cosh(n-1)\theta\over1-2a\cosh\theta+a^2}\tag{714}$$