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In this paper we investigate the finite sum of cosecants$\sum\csc\big(\varphi+a\pi l/n\big),$ where the index $l$ runs through 1 to$n-1$ and $\varphi$ and $a$ are arbitrary parameters, as well as severalclosely related sums, such as similar sums of a series of secants, of tangentsand of cotangents. These trigonometric sums appear in various problems inmathematics, physics, and a variety of related disciplines. Their particularcases were fragmentarily considered in previous works, and it was noted thateven a simple particular case $\sum\csc\big(\pi l/n\big)$ does not have aclosed-form, i.e.~a compact summation formula. In the paper, we derive severalalternative representations for the above-mentioned sums, study theirproperties, relate them to many other finite and infinite sums, obtain theircomplete asymptotic expansions for large $n$ and provide accurate upper andlower bounds (e.g. the typical relative error for the upper bound is lesserthan $2\times10^{-9}$ for $n\geqslant10$ and lesser than $7\times10^{-14}$ for$n\geqslant50$, which is much better than the bounds we could find in previousworks). Our researches reveal that these sums are deeply related to severalspecial numbers and functions, especially to the digamma function (furthermore,as a by-product, we obtain several interesting summations formulae for thedigamma function). Asymptotical studies show that these sums may havequalitatively different behaviour depending on the choice of $\varphi$ and $a$;in particular, as $n$ increases some of them may become sporadically large.Finally, we also provide several historical remarks related to various sumsconsidered in the paper. We show that some results in the field either wererediscovered several times or can easily be deduced from various knownformulae, including some formulae dating back to the XIIXth century.

On a finite sum of cosecants appearing in various problems