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