  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 several
closely related sums, such as similar sums of a series of secants, of tangents
and of cotangents. These trigonometric sums appear in various problems in
mathematics, physics, and a variety of related disciplines. Their particular
cases were fragmentarily considered in previous works, and it was noted that
even a simple particular case $\sum\csc\big(\pi l/n\big)$ does not have a
closed-form, i.e. a compact summation formula. In the paper, we derive several
alternative representations for the above-mentioned sums, study their
properties, relate them to many other finite and infinite sums, obtain their
complete asymptotic expansions for large $n$ and provide accurate upper and
lower bounds (e.g. the typical relative error for the upper bound is lesser
than $2\times10^{-9}$ for $n\geqslant10$ and lesser than $7\times10^{-14}$ for
$n\geqslant50$, which is much better than the bounds we could find in previous
works). Our researches reveal that these sums are deeply related to several
special numbers and functions, especially to the digamma function (furthermore,
as a by-product, we obtain several interesting summations formulae for the
digamma function). Asymptotical studies show that these sums may have
qualitatively 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 sums
considered in the paper. We show that some results in the field either were
rediscovered several times or can easily be deduced from various known
formulae, including some formulae dating back to the XVIIIth century.
