  In this article we develop a general method by which one can explicitly
evaluate certain sums of $n$-th powers of products of $d\geq 1$ elementary
trigonometric functions evaluated at $\mathbf{m}=(m_1,\ldots,m_d)$-th roots of
unity. Our approach is to first identify the individual terms in the expression
under consideration as eigenvalues of a discrete Laplace operator associated to
a graph whose vertices form a $d$-dimensional discrete torus $G_{\mathbf{m}}$
which depends on $\mathbf{m}$. The sums in question are then related to the
$n$-th step of a Markov chain on $G_{\mathbf{m}}$. The Markov chain admits the
interpretation as a particular random walk, also viewed as a discrete time and
discrete space heat diffusion, so then the sum in question is related to
special values of the associated heat kernel. Our evaluation follows by
deriving a combinatorial expression for the heat kernel, which is obtained by
periodizing the heat kernel on the infinite lattice $\mathbb{Z}^{d}$ which
covers $G_{\mathbf{m}}$.
