Trig Stuff 3

$\displaystyle\sum _{p=-5}^5 \left(\displaystyle\sum _{n=-5}^{p-1} \left(\displaystyle\sum _{m=-5}^{n-1} \left(\displaystyle\sum _{k=-5}^{m-1} \left(\displaystyle\sum _{j=-5}^{k-1} \left(\displaystyle\sum _{i=-5}^{j-1} \sin \left(\frac{\pi i}{11}\right) \sin \left(\frac{\pi j}{11}\right) \sin \left(\frac{\pi k}{11}\right) \sin \left(\frac{\pi m}{11}\right) \sin \left(\frac{\pi n}{11}\right) \sin \left(\frac{\pi p}{11}\right)\right)\right)\right)\right)\right) = - \frac{a}{b}$

where a,b are coprime positive integers. Submit $a+b$ .