Question

The value of $\sin \frac{\pi }{10}\sin \frac{13\pi }{10}$ is-

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

Answer 1

The correct option is C $-\frac{1}{4}$

$\sin \frac{\pi }{10}\sin \frac{13\pi }{10}$

$=\sin \frac{\pi }{10}\sin (\pi +\frac{3\pi }{10})$

$=-\sin \frac{\pi }{10}\sin \frac{3\pi }{10}$

We know that $\sin 3A=3\sin A-4{\sin }^{3}A$

$=-\sin \frac{\pi }{10}[3\sin \frac{\pi }{10}-4{\sin }^3\frac{\pi }{10}]$

$=4{\sin }^4\frac{\pi }{10}-3{\sin }^2\frac{\pi }{10}$

$={\sin }^2\frac{\pi }{10}(4{\sin }^2\frac{\pi }{10}-3)$

$={\left(\frac{\sqrt{5}-1}{4}\right)}^2[4{\left(\frac{\sqrt{5}-1}{4}\right)}^2-3]$

$=\frac{5+1-2\sqrt{5}}{16}[4\left(\frac{5+1-2\sqrt{5}}{16}\right)-3]$

$=\frac{6-2\sqrt{5}}{16}[\frac{6-2\sqrt{5}}{4}-3]$

$=\frac{3-\sqrt{5}}{8}\left[\frac{3-\sqrt{5}-6}{2}\right]$

$=\frac{3-\sqrt{5}}{8}\left[\frac{-\sqrt{5}-3}{2}\right]$

$=\frac{(\sqrt{5}-3)(\sqrt{5}+3)}{16}$

$=\frac{5-9}{16}$

$=\frac{-4}{16}=\frac{-1}{4}$