Question

If $\tan \theta +\tan (\theta +\frac{\pi }{3})+\tan (\theta +\frac{2\pi }{3})=3$, then which of the following is equal to $1$?

• A. $\tan 2\theta$
• B. $\tan 3\theta$
• C. ${\tan }^2\theta$
• D. ${\tan }^3\theta$

Answer 1

Answer: (B)

$\tan 3\theta$

Given,
$\tan \theta +\tan (\theta +\frac{\pi }{3})+\tan (\theta +\frac{2\pi }{3})=3$
$\Rightarrow \tan \theta +\frac{\tan \theta +\sqrt{3}}{1-\sqrt{3}\tan \theta }+\frac{\tan \theta -\sqrt{3}}{1+\sqrt{3}\tan \theta }=3$
$\Rightarrow \tan \theta +\frac{8\tan \theta }{1-3{\tan }^2\theta }=3$
$\Rightarrow \frac{9\tan \theta -3{\tan }^2\theta }{1-3{\tan }^2\theta }=3$
$\Rightarrow 3\tan 3\theta =3\Rightarrow \tan 3\theta =1$
Hence, option (B) is correct.