$\begin{array}{l} S = \sin \dfrac{\pi }{{30}}.\sin \dfrac{{7\pi }}{{30}}.\sin \dfrac{{13\pi }}{{30}}.\sin \dfrac{{19\pi }}{{30}}.\sin \dfrac{{25\pi }}{{30}}\\ \cos \dfrac{\pi }{{30}}.S = \sin \dfrac{\pi }{{30}}.\cos \dfrac{{4\pi }}{{15}}.\cos \dfrac{\pi }{{15}}.\cos \left( {\dfrac{{ - 2\pi }}{{15}}} \right).\cos \left( {\dfrac{{ - \pi }}{3}} \right).\cos \dfrac{\pi }{{30}}\\ \cos \dfrac{\pi }{{30}}.S = \sin \dfrac{\pi }{{30}}.\cos \dfrac{\pi }{{30}}.\cos \dfrac{\pi }{{15}}.\cos \dfrac{{2\pi }}{{15}}.\cos \dfrac{{4\pi }}{{15}}.\cos \dfrac{\pi }{3}\\ \cos \dfrac{\pi }{{30}}.S = \dfrac{1}{2}.\sin \dfrac{\pi }{{15}}.\cos \dfrac{\pi }{{15}}.\cos \dfrac{{2\pi }}{{15}}.\cos \dfrac{{4\pi }}{{15}}.\cos \dfrac{\pi }{3}\\ \cos \dfrac{\pi }{{30}}.S = \dfrac{1}{4}.\sin \dfrac{{2\pi }}{{15}}.\cos \dfrac{{2\pi }}{{15}}.\cos \dfrac{{4\pi }}{{15}}.\cos \dfrac{\pi }{3}\\ \cos \dfrac{\pi }{{30}}.S = \dfrac{1}{8}.\sin \dfrac{{4\pi }}{{15}}.\cos \dfrac{{4\pi }}{{15}}.\cos \dfrac{\pi }{3}\\ \cos \dfrac{\pi }{{30}}.S = \dfrac{1}{{16}}.\sin \dfrac{{8\pi }}{{15}}.\cos \dfrac{\pi }{3}\\ \cos \dfrac{\pi }{{30}}.S = \dfrac{1}{{32}}.\cos \left( {\dfrac{{ - \pi }}{{30}}} \right)\\ \Rightarrow S = \dfrac{1}{{32}} \end{array}$
