\(\begin{array}{l}
\sin \frac{\pi }{{16}}.\sin \frac{{3\pi }}{{16}}.\sin \frac{{5\pi }}{{16}}.\sin \frac{{7\pi }}{{16}}\\
 = (\sin \frac{\pi }{{16}}.\sin \frac{{7\pi }}{{16}}).(\sin \frac{{3\pi }}{{16}}.\sin \frac{{5\pi }}{{16}})\\
 = \sin \frac{\pi }{{16}}.\sin (\frac{\pi }{2} - \frac{\pi }{{16}}).\sin \frac{{3\pi }}{{16}}.\sin (\frac{\pi }{2} - \frac{{3\pi }}{{16}})\\
 = \sin \frac{\pi }{{16}}.\cos \frac{\pi }{{16}}.\sin \frac{{3\pi }}{{16}}\cos \frac{{3\pi }}{{16}}\\
 = \frac{1}{2}\sin (\frac{{2\pi }}{{16}}).\frac{1}{2}\sin (\frac{{2.3\pi }}{{16}})\\
 = \frac{1}{4}\sin \frac{\pi }{8}.\sin \frac{{3\pi }}{8}\\
 = \frac{1}{4}.(\frac{{ - 1}}{2}).[\cos (\frac{\pi }{8} + \frac{{3\pi }}{8}) - \cos (\frac{{3\pi }}{8} - \frac{\pi }{8})]\\
 =  - \frac{1}{8}.(\cos \frac{\pi }{2} - \cos \frac{\pi }{4})\\
 = \frac{{ - 1}}{8}.(0 - \frac{{\sqrt 2 }}{2})\\
 = \frac{{\sqrt 2 }}{{16}}
\end{array}\)