Solve this following


If $\tan \left(\frac{\pi}{9}\right), x, \tan \left(\frac{7 \pi}{18}\right) \quad$ are in arithmetic

progression and $\tan \left(\frac{\pi}{9}\right), \mathrm{y}, \tan \left(\frac{5 \pi}{18}\right)$ are also in

arithmetic progression, then $|x-2 y|$ is equal to :


  1. 4

  2. 3

  3. 0

  4. 1

Correct Option: , 3


$x=\frac{1}{2}\left(\tan \frac{\pi}{9}+\tan \frac{7 \pi}{18}\right)$

and $2 \mathrm{y}=\tan \frac{\pi}{9}+\tan \frac{5 \pi}{18}$

so, $x-2 y=\frac{1}{2}\left(\tan \frac{\pi}{9}+\tan \frac{7 \pi}{18}\right)$

$-\left(\tan \frac{\pi}{9}+\tan \frac{5 \pi}{18}\right)$

$\Rightarrow|x-2 y|=\left|\frac{\cot \frac{\pi}{9}-\tan \frac{\pi}{9}}{2}-\tan \frac{5 \pi}{18}\right|$

$=\left|\cot \frac{2 \pi}{9}-\cot \frac{2 \pi}{9}\right|=0$

$\left(\operatorname{astan} \frac{5 \pi}{18}=\cot \frac{2 \pi}{9} ; \tan \frac{7 \pi}{18}=\cot \frac{\pi}{9}\right)$


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