Question

The value of the expression $\tan \frac{\pi }{7}+2\tan \frac{2\pi }{7}+4\tan \frac{4\pi }{7}+8\cot \frac{8\pi }{7}$ is equal to

• A. $cosec\frac{2\pi }{7}+\cot \frac{2\pi }{7}$
• B. $\tan \frac{\pi }{14}-\cot \frac{\pi }{14}$
• C. $\frac{\sin \frac{2\pi }{7}}{1-\cos \frac{2\pi }{7}}$$
• D. $\frac{1+\cos \frac{\pi }{7}+\cos \frac{2\pi }{7}}{\sin \frac{\pi }{7}+\sin \frac{2\pi }{7}}$

Answer 1

The correct option is A $cosec\frac{2\pi }{7}+\cot \frac{2\pi }{7}$

We have

$\tan \frac{\pi }{7}+2\tan \frac{2\pi }{7}+4\tan \frac{4\pi }{7}+8\tan \frac{8\pi }{7}$

$=\tan \left(\pi -\frac{6\pi }{7}\right)+2\tan \left(\pi -\frac{5\pi }{7}\right)+4\tan \frac{4\pi }{7}+8\tan \frac{8\pi }{7}$

$=\cos$$ec$ $\frac{2\pi }{7}+\cos$$ec$$\frac{2\pi }{4}$

Hence, the option $\left(A\right)$ is correct.