Question
The value of $$\sin \frac{\pi }{{14}} \cdot \sin \frac{{3\pi }}{{14}} \cdot \sin \frac{{5\pi }}{{14}} \cdot \sin \frac{{7\pi }}{{14}} \cdot \sin \frac{{9\pi }}{{14}} \cdot \sin \frac{{11\pi }}{{14}} \cdot \sin \frac{{13\pi }}{{14}}$$ is equal to
A.
$$1$$
B.
$$\frac{1}{{16}}$$
C.
$$\frac{1}{{64}}$$
D.
None of these
Answer :
$$\frac{1}{{64}}$$
Solution :
The expression $$ = {\left( {\sin \frac{\pi }{{14}} \cdot \sin \frac{{3\pi }}{{14}}\sin \frac{{5\pi }}{{14}}} \right)^2} \cdot \sin \frac{{7\pi }}{{14}}.$$
Clearly, $$\sin\frac{\pi }{{14}} = \cos \left( {\frac{\pi }{2} - \frac{\pi }{{14}}} \right) = \cos \frac{{3\pi }}{7} = - \cos \left( {x - \frac{{3\pi }}{7}} \right) = - \cos \frac{{4\pi }}{7}.$$
$$\sin\frac{{3\pi }}{{14}} = \cos \frac{{2\pi }}{7},\sin \frac{{5\pi }}{7} = \cos \frac{\pi }{7}.$$
∴ value $$ = {\left( {\cos \frac{\pi }{7}\cos \frac{{2\pi }}{7} \cdot \frac{{4\pi }}{7}} \right)^2} \cdot 1$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\left\{ {\frac{{2\sin \frac{\pi }{7} \cdot \cos \frac{\pi }{7} \cdot \cos \frac{{2\pi }}{7} \cdot \cos \frac{{4\pi }}{7}}}{{2\sin \frac{\pi }{7}}}} \right\}^2} = ..... = \left( {\frac{{\sin \frac{{8\pi }}{7}}}{{8\sin \frac{\pi }{7}}}} \right) = \frac{1}{{{8^2}}}.$$