Question

The value of $tan\theta +tan({60}^{\circ }+\theta )+tan({120}^{\circ }+\theta )$ is

• A. $3tan3\theta$
• B. $tan3\theta$
• C. $3cot3\theta$
• D. $cot3\theta$

Answer 1

The correct option is A

$3tan3\theta$

$tan\theta +tan({60}^{\circ }+\theta )+tan({120}^{\circ }+\theta )=tan\theta +\frac{\sqrt{3}+tan\theta }{1-\sqrt{3}tan\theta }+\frac{(-\sqrt{3}+tan\theta )}{1+\sqrt{3}tan\theta }=\frac{tan\theta (1-3ta{n}^2\theta )+(\sqrt{3}+tan\theta )(1+\sqrt{3}tan\theta )+(-\sqrt{3}+tan\theta )(1-\sqrt{3}tan\theta )}{1-3ta{n}^2\theta }=\frac{tan\theta -3ta{n}^2\theta +\sqrt{3}+3tan\theta +tan\theta +\sqrt{3}ta{n}^2\theta +tan\theta -\sqrt{3}ta{n}^2\theta -\sqrt{3}+3tan\theta }{1-3ta{n}^2\theta }=\frac{9tan\theta -3ta{n}^3\theta }{1-3ta{n}^2\theta }=\frac{3(3tan\theta -ta{n}^2\theta )}{1-3ta{n}^2\theta }=3tan3\theta$