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

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