I'm working on a problem in acoustic scattering, where I have to calculate the torque exerted on a cylinder. The torque, $\tau(\phi,m)$, is a function of the "maximum phase" denoted by $\phi$, and a positive integer $m\ge3$. After some algebraic manipulation, $\tau(\phi,m)$ reduces to the following equation: $$\tau(\phi,m)=\sum_{i=1}^{\left\lfloor\frac{m}{2}\right\rfloor}\sin\left(\frac{2\pi i}{m}\right)\left[(m-i)\sin(i\phi)-i\sin((m-i)\phi)\right]$$ Plotting $\tau$ for several $m$s, It seems that the function is non-negative for $0\le \phi\le\pi$; i.e. $$\tau(\phi,m)\ge 0 \>\>\>\>\>\> 0\le \phi\le\pi, \>\> m\ge3$$ Is there an easy proof for it?
So far, I've found the zeros of $\tau$, but haven't been able to show they're the "only" zeros of the function. I also tried to transform the above identity into a sum of ChebyshevU polynomials and use the inequalities associated with them, but without success. Any help would be appreciated.