Question

tan 20° + tan 40° + $\sqrt{3}$ tan 20° tan 40° is equal to

(a) $\frac{\sqrt{3}}{4}$
(b) $\frac{\sqrt{3}}{2}$
(c) $\sqrt{3}$
(d) 1

Answer 1

(c) $\sqrt{3}$

$$\begin{gathered} \tan 20°+\tan 40°+\sqrt{3}\tan 20°\tan 40° \\ =\tan 60°(1-\tan 20°\tan 40°)+\tan 60°\tan 20°\tan 40°\left[\mathrm{Using}\tan 60°=\frac{\tan 20+\tan 40}{1-\tan 20\tan 40}\mathrm{and}\tan 60°=\sqrt{3}\right] \\ =\tan 60°-\tan 60°\tan 20°\tan 40°+\tan 60°\tan 20°\tan 40° \\ =\tan 60° \\ =\sqrt{3} \end{gathered}$$