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 $\tan 40^{\circ }+\tan 20^{\circ }+\sqrt{3}\tan 40^{\circ }\tan 20^{\circ }$
$=\sqrt{3}$