Question

For $0<\theta <\pi /2,tan\theta +tan2\theta +tan3\theta =0$ if

• A. $tan\theta =0$
• B. $tan2\theta =0$
• C. $tan3\theta =0$
• D. $tan\theta tan2\theta =2$

Answer 1

Answer: (C), (D)

The correct options are
C $tan3\theta =0$
D $tan\theta tan2\theta =2$

$tan\theta +tan2\theta +tan3\theta =0$

$\Rightarrow tan\theta +tan2\theta =-tan3\theta$ ...(1)

Now,

$tan3\theta =tan(\theta +2\theta )=\frac{tan\theta +tan2\theta }{1-tan\theta tan2\theta }$

$\Rightarrow tan3\theta (1-tan\theta tan2\theta )=tan\theta +tan2\theta =-tan3\theta$

$\Rightarrow tan3\theta (2-tan\theta tan2\theta )=0$

We get

$tan3\theta =0$ or $tan\theta tan2\theta =2$