The value of

$2 \sin \left(\frac{\pi}{8}\right) \sin \left(\frac{2 \pi}{8}\right) \sin \left(\frac{3 \pi}{8}\right) \sin \left(\frac{5 \pi}{8}\right) \sin \left(\frac{6 \pi}{8}\right) \sin \left(\frac{7 \pi}{8}\right)$

$2 \sin ^{2} \frac{\pi}{8} \sin ^{2} \frac{2 \pi}{8} \sin ^{2} \frac{3 \pi}{8}$

$\sin ^{2} \frac{\pi}{8} \sin ^{2} \frac{3 \pi}{8}$

$\sin ^{2} \frac{\pi}{8} \cos ^{2} \frac{\pi}{8}$

$\frac{1}{4} \sin ^{2}\left(\frac{\pi}{4}\right)=\frac{1}{8}$