Question

The value of $2\sin \left(\frac{\pi }{8}\right)\sin \left(\frac{2\pi }{8}\right)\sin \left(\frac{3\pi }{8}\right)\sin \left(\frac{5\pi }{8}\right)\sin \left(\frac{6\pi }{8}\right)\sin \left(\frac{7\pi }{8}\right)$ is

• A. $\frac{1}{8\sqrt{2}}$
• B. $\frac{1}{4\sqrt{2}}$
• C. $\frac{1}{8}$
• D. $\frac{1}{4}$

Answer 1

The correct option is C $\frac{1}{8}$

We know
$\sin \frac{5\pi }{8}=\sin (\pi -\frac{3\pi }{8})=\sin \left(\frac{3\pi }{8}\right)\cdots \left(i\right)$
$\sin \frac{7\pi }{8}=\sin (\pi -\frac{\pi }{8})=\sin \frac{\pi }{8}\cdots \left(ii\right)$
$\sin \frac{2\pi }{8}=\sin \frac{\pi }{4}=\frac{1}{\sqrt{2}}\cdots \left(iii\right)$
$\sin \frac{6\pi }{8}=\sin \frac{3\pi }{4}=\frac{1}{\sqrt{2}}\cdots \left(iv\right)$

$2\sin \left(\frac{\pi }{8}\right)\sin \left(\frac{2\pi }{8}\right)\sin \left(\frac{3\pi }{8}\right)\sin \left(\frac{5\pi }{8}\right)\sin \left(\frac{6\pi }{8}\right)\sin \left(\frac{7\pi }{8}\right)$
$=2{\sin }^2\frac{\pi }{8}\times {\sin }^2\left(\frac{3\pi }{8}\right)\times \left(\frac{1}{\sqrt{2}}\right)\times \left(\frac{1}{\sqrt{2}}\right)$
$=2{\sin }^2\frac{\pi }{8}\times {\sin }^2\left(\frac{3\pi }{8}\right)\times \frac{1}{2}$
$={(\sin \frac{\pi }{8}.\cos \frac{\pi }{8})}^2[\because \sin \frac{3\pi }{8}=\sin (\frac{\pi }{2}-\frac{\pi }{8})=\cos \frac{\pi }{8}]$
$={\left(\frac{1}{2}\sin \frac{\pi }{4}\right)}^2={\left(\frac{1}{2\sqrt{2}}\right)}^2=\frac{1}{8}$