Calculate $\int_0^{2\pi}{(\cos\theta+i\sin\theta\cos\phi)^l \cdot 2i\sin(m\phi)\,d\phi}$

Question

I want to evaluate $$ \int_0^{2\pi} \left[\cos\left(\theta\right) + {\rm i}\sin\left(\theta\right)\cos\left(\phi\right)\right]^{\,\ell}\ 2{\rm i}\sin\left(m\phi\right)\,{\rm d}\phi\ \mbox{where}\ \ell,m\in\mathbb{Z}^{+} $$ If i put values for $\ell$ and $m$ like $1$ and $2$ (for example), the integral is zero when I evaluate this in a calculator so i suspect the integral is zero: I tried to prove this but since there is a polynomial of degree $\ell$ and the sine function is evaluated in $m\phi$, I failed to do this. I appreciate any suggestions.