  We prove several old and new theorems about finite sums involving characters
and trigonometric functions. These sums can be traced back to theta function
identities from Ramanujan's notebooks and were systematically first studied by
Berndt and Zaharescu; their proofs involved complex contour integration. We
show how to prove most of Berndt-Zaharescu's and some new identities by
elementary methods of discrete Fourier Analysis.
