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. $0$
• B. $1$
• C. $3$
• D. $4$

Answer 1

The correct option is A $0$

$x-2y=\frac{\tan {20}^{\circ }+\tan {70}^{\circ }}{2}-(\tan {20}^{\circ }+\tan {50}^{\circ })$
$\frac{1}{2}(\tan {70}^{\circ }-\tan {20}^{\circ }-2\tan {50}^{\circ })$
$=\frac{1}{2}\left[\right(\tan {70}^{\circ }-\tan {50}^{\circ })-(\tan {20}^{\circ }+\tan {50}^{\circ }\left)\right]$
$=\frac{1}{2}[\frac{\sin {20}^{\circ }}{\cos {70}^{\circ }\cos {50}^{\circ }}-\frac{\sin {70}^{\circ }}{\cos {20}^{\circ }\cos {50}^{\circ }}]$
$=\frac{1}{2}[\frac{1}{\cos {50}^{\circ }}-\frac{1}{\cos {50}^{\circ }}]=0$