Question

The value of $ tan\text{ }40{}^\circ +tan\text{ }20{}^\circ +\sqrt{3}\text{ }tan\text{ }20{}^\circ tan\text{ }40{}^\circ $ is equal to

• A. $ \sqrt{12} $
• B. $ \frac{1}{\sqrt{3}} $
• C. 1
• D. $ \frac{\sqrt{3}}{2} $
• E. $ \sqrt{3} $

Answer 1

Answer: (E)

We know that, $ tan\text{(}20{}^\circ +40{}^\circ )=\frac{\tan 20{}^\circ +\tan 40{}^\circ }{1-\tan 20{}^\circ \tan 40{}^\circ } $
$ \Rightarrow $ $ \sqrt{3}=\frac{\tan 20{}^\circ +\tan 40{}^\circ }{1-\tan 20{}^\circ \tan 40{}^\circ } $
$ \Rightarrow $ $ \sqrt{3}-\sqrt{3}\tan 20{}^\circ \tan 40{}^\circ $
$=\tan 20{}^\circ +\tan 40{}^\circ $
$ \Rightarrow $ $ \tan 20{}^\circ +\tan 40{}^\circ +\sqrt{3}\tan 20{}^\circ \tan 40{}^\circ $
$=\sqrt{3} $