Question

$cosec\frac{{360}^o}{7}+cosec\frac{{540}^o}{7}=$

• A. $cosec\frac{{180}^o}{7}$
• B. $cosec\frac{{90}^o}{7}$
• C. $\sec \frac{{180}^o}{7}$
• D. $\sec \frac{{90}^o}{7}$

Answer 1

Answer: (A)

$cosec\frac{{180}^o}{7}$

$\csc \frac{2\pi }{7}+\csc \frac{3\pi }{7}=\frac{1}{\sin \frac{2\pi }{7}}+\frac{1}{\sin \frac{3\pi }{7}}=\frac{\sin \frac{3\pi }{7}+\sin \frac{2\pi }{7}}{\sin \frac{2\pi }{7}.\sin \frac{3\pi }{7}}=\frac{\sin \frac{3\pi }{7}+\sin (\pi -\frac{5\pi }{7})}{\sin \frac{2\pi }{7}.\sin \frac{3\pi }{7}}\therefore \csc \frac{2\pi }{7}+\csc \frac{3\pi }{7}=\frac{\sin \frac{5\pi }{7}+\sin \frac{3\pi }{7}}{\sin \frac{2\pi }{7}.\sin (\pi -\frac{4\pi }{7})}=\frac{2\sin \frac{4\pi }{7}.\cos \frac{\pi }{7}}{\sin \frac{2\pi }{7}.\sin \frac{4\pi }{7}}=2.\frac{\cos \frac{\pi }{7}}{2.\sin \frac{\pi }{7}.\cos \frac{\pi }{7}}=\csc \frac{\pi }{7}\csc \frac{2\pi }{7}+\csc \frac{3\pi }{7}=\csc \frac{\pi }{7}=\csc \frac{{180}^{\circ }}{7}$