  This paper presents a combinatorial study of sums of integer powers of the
cotangent which is a popular theme in classical calculus. Our main tool the
realization of cotangent values as eigenvalues of a simple self-adjoint matrix
with integer matrix. We use the trace method to draw conclusions about integer
values of the sums and expand generating functions to obtain explicit
evaluations. It is remarkable that throughout the calculations the
combinatorics are governed by the higher tangent and arctangent numbers
exclusively. Finally we indicate a new approximation of the values of the
Riemann zeta function at even integer arguments.
