Question

The positive integer value of $n>3$ satisfying the equation $\frac{1}{\sin \left(\frac{\pi }{n}\right)}=\frac{1}{\sin \left(\frac{2\pi }{n}\right)}+\frac{1}{\sin \left(\frac{3\pi }{n}\right)}$ is____

Answer 1

Answer: 7

$\frac{1}{\sin \frac{\pi }{n}}-\frac{1}{\sin \frac{3\pi }{n}}=\frac{1}{\sin \frac{2\pi }{n}}$
$\Rightarrow \frac{\sin \frac{3\pi }{n}-\sin \frac{\pi }{n}}{\sin \frac{\pi }{n}\sin \frac{3\pi }{n}}=\frac{1}{\sin \frac{2\pi }{n}}\frac{\left(2\sin \frac{\pi }{n}\cos \frac{2\pi }{n}\right)\sin \frac{2\pi }{n}}{\sin \frac{\pi }{n}\sin \frac{3\pi }{n}}=1$
$\Rightarrow \sin \frac{4\pi }{n}=\sin \frac{3\pi }{n}$
$\Rightarrow \frac{4\pi }{n}+\frac{3\pi }{n}=\pi$
$\Rightarrow n=7$