After working with a Fourier series for a while, I realized that it would be of great help to me if I could prove that the following limit is zero: $$\lim_{N\to\infty}\frac{1}{N}\sum_{m=1}^\infty\frac{(-1)^m\cos(m(N+1))\sin(mN)}{(4m^2-1)\sin(m)}$$

I tried to do several things and I can show that with a trivial upper bound that if $\sum_{m=1}^\infty\frac{1}{m^2|\sin(m)|}$ converges then the limit will clearly be zero, but this only seems to complicate things, as I think I would have to introduce notions of Diophantine approximation to show that the denominator doesn't get too crazy.