Q. If $\tan{\theta + \tan\left( \theta + \frac{\pi}{3} \right) + \tan\left( \theta + \frac{2\pi}{3} \right)} = 3$, then which of the following is equal to 1?
A) $\tan{2\theta}$
B) $\tan{3\theta}$
C) $\tan^{2}\theta$
D) $\tan^{3}\theta$
Explanation
(b)
$\tan{\theta + \frac{\tan{\theta + \sqrt{3}}}{1 - \sqrt{3}\tan\theta} + \frac{\tan{\theta - \sqrt{3}}}{1 + \sqrt{3}\tan\theta}} = 3$
$\Rightarrow \ \tan{\theta + \frac{8\tan\theta}{1 - 3\tan^{2}\theta} = 3}$
$\Rightarrow \ \ \frac{9\tan{\theta - 3\tan^{3}\theta}}{1 - 3\tan^{2}\theta} = 3$
$\Rightarrow \ \ \ \ \ \ 3\tan{3\theta = 3}\ \ \ \Rightarrow \ \ \ \tan{3\theta = 1}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.