  Let $X_m$ denote the discrete circle with $m$ vertices. For $x,y\in X_{m}$
and complex $s$, let $G_{X_m,\chi_{\beta}}(x,y;s)$ be the resolvent kernel
associated to the combinatorial Laplacian which acts on the space of functions
on $X_{m}$ that are twisted by a character $\chi_{\beta}$. We will compute
$G_{X_m,\chi_{\beta}}(x,y;s)$ in two different ways. First, using the spectral
expansion of the Laplacian, we show that $G_{X_m,\chi_{\beta}}(x,y;s)$ is a
generating function for certain trigonometric sums involving powers of the
cosecant function; by choosing $\beta$ or $s$ appropriately, the sums in
question involve powers of the secant function. Second, by viewing $X_{m}$ as a
quotient space of $\mathbb{Z}$, we prove that $G_{X_m,\chi_{\beta}}(x,y;s)$ is
a rational function which is given in terms of Chebyshev polynomials. From the
existence and uniqueness of $G_{X_m,\chi_{\beta}}(x,y;s)$, these two
evaluations are equal. From the resulting identity, we obtain a means by which
one can obtain explicit evaluations of cosecant and secant sums. The identities
we prove depend on a number of parameters, and when we specialize the values of
these parameters we obtain several previously known formulas. Going further, we
derive a recursion formula for special values of the $L$-functions associated
to the cycle graph $X_{m}$, thus answering a question from arXiv:2212.13687v1.
