$2 \sin \frac{\pi}{22} \sin \frac{3 \pi}{22} \sin \frac{5 \pi}{22} \sin \frac{7 \pi}{22} \sin \frac{9 \pi}{22}$

$=2 \sin \left(\frac{11 \pi-10 \pi}{22}\right) \sin \left(\frac{11 \pi-8 \pi}{22}\right) \sin \left(\frac{11 \pi-6 \pi}{22}\right)$ $\sin \left(\frac{11 \pi-4 \pi}{22}\right) \sin \left(\frac{11 \pi-2 \pi}{22}\right)$

$=2 \cos \frac{\pi}{11} \cos \frac{2 \pi}{11} \cos \frac{3 \pi}{11} \cos \frac{4 \pi}{11} \cos \frac{5 \pi}{11}$

$=\frac{2 \sin \frac{32 \pi}{11}}{2^{5} \sin \frac{\pi}{11}}$

$=\frac{1}{16}$