$2\sin \left(\frac{\pi }{22}\right)\sin \left(\frac{3\pi }{22}\right)\sin \left(\frac{5\pi }{22}\right)\sin \left(\frac{7\pi }{22}\right)\sin \left(\frac{9\pi }{22}\right)$ is equal to

$\begin{array}{l}2\sin \left(\frac{\pi }{22}\right)\sin \left(\frac{3\pi }{22}\right)\sin \left(\frac{5\pi }{22}\right)\sin \left(\frac{7\pi }{22}\right)\sin \left(\frac{9\pi }{22}\right)\\ =2\cos (\frac{\pi }{2}-\frac{\pi }{22})\cos (\frac{\pi }{2}-\frac{3\pi }{22})\cos (\frac{\pi }{2}-\frac{5\pi }{22})\cos (\frac{\pi }{2}-\frac{7\pi }{22})\cos (\frac{\pi }{2}-\frac{9\pi }{22})\end{array}$

$\begin{array}{l}=2\cos \left(\frac{10\pi }{22}\right)\cos \left(\frac{8\pi }{22}\right)\cos \left(\frac{6\pi }{22}\right)\cos \left(\frac{4\pi }{22}\right)\cos \left(\frac{2\pi }{22}\right)\\ =2\cos \left(\frac{\pi }{11}\right)\cos \left(\frac{2\pi }{11}\right)\cos \left(\frac{3\pi }{11}\right)\cos \left(\frac{4\pi }{11}\right)\cos \left(\frac{5\pi }{11}\right)\\ =2\cos \left(\frac{\pi }{11}\right)\cos \left(\frac{2\pi }{11}\right)\cos \left(\frac{4\pi }{11}\right)\cos (\pi -\frac{3\pi }{11})\cos (\pi +\frac{5\pi }{11})\\ =2\cos \left(\frac{\pi }{11}\right)\cos \left(\frac{2\pi }{11}\right)\cos \left(\frac{4\pi }{11}\right)\cos \left(\frac{8\pi }{11}\right)\cos \left(\frac{16\pi }{11}\right)\end{array}$

$=\frac{2.\sin (2^5\times \frac{\pi }{11})}{2^5\sin \frac{\pi }{11}}=\frac{2.\sin \left(\frac{32\pi }{11}\right)}{32\sin \frac{\pi }{11}}=\frac{1}{16}$