Some series over the product of two trigonometric functions and series involving Bessel functions

The sum of the series [GRAPHICS] involving the product of two trigonometric functions is obtained using the sum of the series [GRAPHICS] whose terms involve one trigonometric function. The first series is represented as series in terms of the Riemann zeta and related functions, which has a closed form in certain cases. Some applications of these results to the summation of series containing Bessel functions are given. The obtained results also include as special cases formulas in some known books. We further show how to make use of these results to obtain closed form solutions of some boundary value problems in mathematical physics.