The value of 2 sin π 8 sin 2 π 8 sin 3 π 8 sin 5 π 8 sin 6 π 8 sin 7 π 8 is :

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

$\Rightarrow 2\sin \left(\frac{\mathrm{\pi }}{8}\right)\sin \left(\frac{2\mathrm{\pi }}{8}\right)\sin \left(\frac{3\mathrm{\pi }}{8}\right)\sin \left(\mathrm{\pi }-\frac{3\mathrm{\pi }}{8}\right)\sin \left(\mathrm{\pi }-\frac{2\mathrm{\pi }}{8}\right)\sin \left(\mathrm{\pi }-\frac{\mathrm{\pi }}{8}\right)$

$\Rightarrow 2{\sin }^2\frac{\mathrm{\pi }}{8}{\sin }^2\left(\frac{2\mathrm{\pi }}{8}\right){\sin }^2\left(\frac{3\mathrm{\pi }}{8}\right)$

$\Rightarrow 2{\sin }^2\frac{\mathrm{\pi }}{8}{\sin }^2\left(\frac{2\mathrm{\pi }}{8}\right){\sin }^2\left(\frac{3\mathrm{\pi }}{8}\right)$

$\Rightarrow 2{\sin }^2\frac{\mathrm{\pi }}{8}{\sin }^2\left(\frac{\mathrm{\pi }}{4}\right){\sin }^2\left(\frac{3\mathrm{\pi }}{8}\right)$

$\Rightarrow {\sin }^2\frac{\mathrm{\pi }}{8}{\sin }^2\left(\frac{3\mathrm{\pi }}{8}\right)$

$\Rightarrow {\sin }^2\frac{\mathrm{\pi }}{8}{\sin }^2\left(\frac{\mathrm{\pi }}{2}-\frac{\mathrm{\pi }}{8}\right)$

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

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

$\frac{1}{4}{\sin }^2\frac{\mathrm{\pi }}{4}=\frac{1}{8}$