Question

If $\tan \left(\frac{\pi }{9}\right),x,\tan \left(\frac{7\pi }{18}\right)$ are in arithmetic progression and $\tan \left(\frac{\pi }{9}\right),y,\tan \left(\frac{5\pi }{18}\right)$ are also in arithmetic progression, then $|x-2y|$ is equal to :

• A. 4
• B. 3
• C. 0
• D. 1

Answer 1

Answer: (C)

$x=\frac{1}{2}\left(\tan \frac{\pi }{9}+\tan \frac{7\pi }{18}\right)$
and $2y=\tan \frac{\pi }{9}+\tan \frac{5\pi }{18}$
SO, $x-2y=\frac{1}{2}\left(\tan \frac{\pi }{9}+\tan \frac{7\pi }{18}\right)$
$-\left(\tan \frac{\pi }{9}+\tan \frac{5\pi }{18}\right)$
$\Rightarrow |x-2y|=\left|\frac{\cot \frac{\pi }{9}-\tan \frac{\pi }{9}}{2}-\tan \frac{5\pi }{18}\right|$
$=\left|\cot \frac{2\pi }{9}-\cot \frac{2\pi }{9}\right|=0$
$($ as $\tan \frac{5\pi }{18}=\cot \frac{2\pi }{9};\tan \frac{7\pi }{18}=\cot \frac{\pi }{9})$