Two other references to similar sums are 

Bruce C. Berndt and Boon Pin Yeap, [Explicit evaluations and reciprocity theorems for finite trigonometric sums][1], Advances in Applied Mathematics
Volume 29, Issue 3, October 2002, Pages 358--385

and 

Ira Gessel, [Generating Functions and Generalized Dedekind Sums][2], Electronic J. Combinatorics, 
Volume 4, Issue 2 (1997) (The Wilf Festschrift volume), R11.

The paper of Berndt and Yeap uses contour integration and has an extensive list of references. My paper uses elementary methods, including partial fractions. 


  [1]: http://www.math.uiuc.edu/~berndt/articles/trigsums.pdf
  [2]: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v4i2r11