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 :
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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