Question

The value of $sin \frac{\pi}{14}.sin\frac{3\pi}{14}.sin\frac{5\pi}{14}.sin\frac{7\pi}{14}.sin\frac{9\pi}{14}.sin\frac{11\pi}{14}.sin\frac{13\pi}{14}$ is equal to ............

Answer 1

$sin \frac{\pi}{14}.sin\frac{3\pi}{14}.sin\frac{5\pi}{14}.sin\frac{7\pi}{14}.sin\frac{9\pi}{14}.sin\frac{11\pi}{14}.sin\frac{13\pi}{14}$
$=sin\frac{\pi}{14}.sin\frac{3\pi}{14}.sin\frac{5\pi}{14}.sin\Bigg(\pi-\frac{5\pi}{14}\Bigg).sin\Bigg(\pi-\frac{3\pi}{14}\Bigg).sin\Bigg(\pi-\frac{\pi}{14}\Bigg)$
$=sin^2\frac{\pi}{14}.sin^2\frac{3\pi}{14}.sin^2\frac{5\pi}{14}=\Bigg(sin\frac{5\pi}{14}.sin\frac{3\pi}{14}.sin\frac{\pi}{14}\Bigg)$
$=\Bigg(cos\Bigg(\frac{\pi}{2}-\frac{\pi}{14}\Bigg).cos\Bigg(\frac{\pi}{2}-\frac{3\pi}{14}\Bigg).cos\Bigg(\frac{\pi}{2}-\frac{5\pi}{14}\Bigg)\Bigg)^2$
$=\Bigg(cos\frac{3\pi}{7}.cos\frac{2\pi}{7}.cos\frac{\pi}{7}\Bigg)^2$
$=\Bigg(-cos\frac{\pi}{7}.cos\frac{2\pi}{7}.cos\frac{\pi}{7}\Bigg)^2$
$=\Bigg(-cos\frac{\pi}{7}.cos\frac{2\pi}{7}.cos\frac{4\pi}{7}\Bigg)^2=\Bigg(\frac{sin 2^3 \pi/7}{2^3. sin \pi/7}\Bigg)^2$
$=\Bigg(-\frac{1}{8}.\frac{sin 8\pi/7}{sin \pi/7}\Bigg)^2\Bigg[\because sin \frac{8\pi}{7}=sin\Bigg(\Bigg(\pi+\frac{\pi}{7}\Bigg)=-sin\frac{\pi}{7}\Bigg]$
$1/64.$