Question

$\tan 20^{\circ}+\tan 40^{\circ}+3 \tan 20^{\circ} \tan 40^{\circ}$ is equal to

• A. $\sqrt{3} / 2$
• B. $\sqrt{3} / 4$
• C. $\sqrt{3} $
• D. $1$

Answer 1

Answer: (C)

$\sqrt{3}=\tan 60^{\circ}=\tan \left(40^{\circ}+20^{\circ}\right)$
$=\frac{\tan 40^{\circ}+\tan 20^{\circ}}{1-\tan 40^{\circ} \tan 20^{\circ}}$
$\therefore \sqrt{3}-\sqrt{3} \tan 40^{\circ} \tan 20^{\circ}$
$=\tan 40^{\circ}+\tan 20^{\circ}$
Hence $\tan 40^{\circ}+\tan 20^{\circ}+\sqrt{3} \tan 40^{\circ} \tan 20^{\circ}$
$=\sqrt{3}$