An Elementary Derivation of Finite Cotangent Sums

Finite cotangent sums are closely related to values of the Riemann zeta function at the positive even integers. Over the past decades, numerous authors have obtained explicit evaluations of these sums. In this note, we present an intuitive evaluation using Laurent coefficients of powers of the cotangent function. In fact, our expression is a concise restatement of a well-known formula by Berndt and Yeap that was obtained using contour integration.