Answer:
Option C
Explanation:
Given ,n>3 $\epsilon$ integer
and $\frac{1}{\sin\left(\frac{\pi}{n}\right)}=\frac{1}{\sin\left(\frac{2\pi}{n}\right)}+\frac{1}{\sin\left(\frac{3\pi}{n}\right)}$
$\Rightarrow$ $\frac{1}{\sin\frac{\pi}{n}}-\frac{1}{\sin\frac{3\pi}{n}}=\frac{1}{\sin\frac{2\pi}{n}}$
$\Rightarrow$ $\frac{\sin\frac{3\pi}{n}-\sin\frac{\pi}{n}}{\sin\frac{\pi}{n}.\sin\frac{3\pi}{n}}=\frac{1}{\sin\frac{2\pi}{n}}$
$\Rightarrow$ $2 \cos \left(\frac{2 \pi}{n}\right).\sin \frac{\pi}{n}=\frac{\sin\frac{\pi}{n}.\sin\frac{3\pi}{n}}{\sin\frac{2\pi}{n}}$
$\Rightarrow$ $2 \sin \frac{2 \pi}{n}.\cos \frac{2\pi}{n}= \sin \frac{3 \pi}{n}$
$\Rightarrow$ $\sin \frac{4 \pi}{3}=\sin \frac{ 3 \pi}{n}$
$\Rightarrow$ $\frac{ 4 \pi}{n}= \pi-\frac{3 \pi}{n} \Rightarrow \frac{7 \pi}{n} = \pi \Rightarrow n=7$
