Consider $k,l,d\in \mathbb{Z}$ such that $0<k<l<d$ and take $\alpha=\frac{2\pi k}{d}$ and $\beta=\frac{2\pi l}{d}$. I need to prove that $$\sum_{n=1}^\infty\frac{sin(n\alpha)-\sin(n\beta)+\sin(n(\beta-\alpha))}{n^2}>0$$
Motivation: I'm interested in computing the values of a certain function $V:\mathbb{C}\times\mathbb{C}\rightarrow \mathbb{R}$ (a volume function coming from number theory) when $(x,y)$ are the roots of unity. What is written above is the value at $x=e^{\frac{2\pi k}{d}i}$ and $y=e^{\frac{2\pi l}{d}i}$ and if I prove this result it will have some application on my specific problem.
I tried to work with a concrete formula for $\sum_{n=1}^\infty \frac{\sin(nx)}{n^2}$ but it appears that this series is particularly difficult as it is written here: https://arxiv.org/pdf/1308.2626.pdf.
I also tried to use that $$\sin(x)-\sin(y)+\sin(y-x)=-4 \sin(x/2) \sin(x/2 - y/2) \sin(y/2)$$ but I couldn't do much more.