If `tantheta + tan(theta + pi/3) + tan(theta-pi/3)= Ktan3theta`, then K is equal to

Options

A. 1
B. 3
C. `1/3`
D. none of these

Answer

B

Explanation

We have,
`tan theta+tan(theta+(pi)/(3))+tan(theta-(pi)/(3))`
`=tan theta+(tantheta+sqrt3)/(1-sqrt3tantheta)+(tantheta-sqrt3)/(1+sqrt3tantheta)`
`tan theta(1-3tan^(2)theta)+(tan theta+sqrt3)(1+sqrt3tantheta)`
` =(+(tantheta-sqrt3)(1-sqrt3tantheta))/(1-3tan^(2)theta)`
`=3((3tantheta-tan^(3)theta)/(1-3tan^(2)theta))=3tan3 theta`
`therefore k=3.`