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

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