Solving problem with basic trigonometry formula and double angle formula

$\displaystyle \text{Prove}\ \text{that}\ \operatorname{cosec}\frac{{{{{180}}^{\circ }}}}{7}=\ \operatorname{cosec}\frac{{{{{360}}^{\circ }}}}{7}+\ \operatorname{cosec}\frac{{{{{540}}^{\circ }}}}{7}.$


Solution

$\displaystyle \ \operatorname{cosec}\frac{{{{{360}}^{\circ }}}}{7}+\ \operatorname{cosec}\frac{{{{{540}}^{\circ }}}}{7}=\frac{1}{{\sin \frac{{2\pi }}{7}}}+\frac{1}{{\sin \frac{{3\pi }}{7}}}$

$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{{\sin \frac{{3\pi }}{7}+\sin \frac{{2\pi }}{7}}}{{\sin \frac{{2\pi }}{7}\sin \frac{{3\pi }}{7}}}$

$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{{2\sin \frac{{5\pi }}{{14}}\cos \frac{\pi }{{14}}}}{{2\sin \frac{\pi }{7}\cos \frac{\pi }{7}\cos \frac{\pi }{{14}}}}$

$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{{\cos \frac{\pi }{7}}}{{\sin \frac{\pi }{7}\cos \frac{\pi }{7}}}$ 

$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\ \operatorname{cosec}\frac{\pi }{7}$

$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\ \operatorname{cosec}\frac{{{{{180}}^{\circ }}}}{7}$