  We present a new and elegant integral approach to computing the
Gardner-Fisher trigonometric power sum, which is given by $$
S_{m,v}=\left(\frac
\pi{2m}\right)^{2v}\sum_{k=1}^{m-1}\cos^{-2v}\left(\frac{k\pi}{2m}\right)\, ,
$$ We present a new and elegant integral approach to computing the
Gardner-Fisher trigonometric power sum, which is given by $$
S_{m,v}=\left(\frac
\pi{2m}\right)^{2v}\sum_{k=1}^{m-1}\cos^{-2v}\left(\frac{k\pi}{2m}\right)\, ,
$$ where $m$ and $v$ are positive integers. This method not only confirms the
results obtained earlier by an empirical method, but it is also much more
expedient from a computational point of view. By comparing the formulas from
both methods, we derive several new interesting number theoretic results
involving symmetric polynomials over the set of quadratic powers up to
$(v-1)^2$ and the generalized cosecant numbers. The method is then extended to
other related trigonometric power sums including the untwisted Dowker sum. By
comparing both forms for this important sum, we derive new formulas for
specific values of the N\"{o}rlund polynomials. Finally, by using the results
appearing in the tables, we consider more advanced sums involving the product
of powers of cotangent and tangent with powers of cosecant and secant
respectively.
