Mathematics - Trigonometric Equation Question with Solution | TestHub

Given, $n>3\in$ Integer $$\begin{array}{rl}& \text{ and }\frac{1}{\sin \left(\frac{\pi }{n}\right)}=\frac{1}{\sin \left(\frac{2\pi }{n}\right)}+\frac{1}{\sin \left(\frac{3\pi }{n}\right)}\\ & \Rightarrow \frac{1}{\sin \frac{\pi }{n}}-\frac{1}{\sin \frac{3\pi }{n}}=\frac{1}{\sin \frac{2\pi }{n}}\\ & \Rightarrow \frac{\sin \frac{3\pi }{n}-\sin \frac{\pi }{n}}{\sin \frac{\pi }{n}\cdot \sin \frac{3\pi }{n}}=\frac{1}{\sin \frac{2\pi }{n}}\end{array}$$ $$\begin{array}{rl}& \Rightarrow 2\cos \left(\frac{2\pi }{n}\right)\cdot \sin \frac{\pi }{n}=\frac{\sin \frac{\pi }{n}\cdot \sin \frac{3\pi }{n}}{\sin \frac{2\pi }{n}}\\ & \Rightarrow 2\sin \frac{2\pi }{n}\cdot \cos \frac{2\pi }{n}=\sin \frac{3\pi }{n}\\ & \Rightarrow \sin \frac{4\pi }{n}=\sin \frac{3\pi }{n}\\ & \Rightarrow \frac{4\pi }{n}=\pi -\frac{3\pi }{n}\Rightarrow \frac{7\pi }{n}=\pi \Rightarrow n=7\end{array}$$