$2\sin \left( {{\pi \over 8}} \right)\sin \left( {{{2\pi } \over 8}} \right)\sin \left( {{{3\pi } \over 8}} \right)\sin \left( {{{5\pi } \over 8}} \right)\sin \left( {{{6\pi } \over 8}} \right)\sin \left( {{{7\pi } \over 8}} \right)$ $2{\sin ^2}{\pi \over 8}{\sin ^2}{{2\pi } \over 8}{\sin ^2}{{3\pi } \over 8}$ ${\sin ^2}{\pi \over 8}{\sin ^2}{{3\pi } \over 8}$ ${\sin ^2}{\pi \over 8}{\cos ^2}{\pi \over 8}$ ${1 \over 4}{\sin ^2}\left( {{\pi \over 4}} \right) = {1 \over 8}$