Question

Given that $\tan \, \theta = m \neq 0, \tan \, 2 \theta = n \neq 0$ and $\tan \, \theta + \tan \, 2 \theta = \tan \, 3 \theta$, then which one of the following is correct ?

• A. m = n
• B. m + n = 1
• C. m + n = 0
• D. mn = - 1

Answer 1

Answer: (C)

m + n = 0

Answer 2

Given that $\tan \, \theta = m$ and $\tan \, 2 \theta = n$ We know from fundamentals that $\Rightarrow \tan3\theta = \frac{\tan\theta+\tan2\theta}{1-\tan\theta \tan2\theta}$ Since , $\tan3 \theta = \tan\theta+\tan2 \theta .....$ as given) $\Rightarrow \tan\theta + \tan2 \theta = \frac{\tan\theta+\tan2\theta}{1-\tan\theta \tan2 \theta}$ $\Rightarrow \left(\tan\theta + \tan2\theta\right)\left(1- \tan\theta \tan2 \theta\right) - \left(\tan\theta + \tan2 \theta\right) = 0$ \Rightarrow \left(\tan\theta + \tan2\theta\right)\left\\{1-\tan\theta \tan2 \theta-1\right\\} = 0 $\Rightarrow \left(\tan\theta + \tan2\theta \right)\left(\tan\theta \tan2\theta \right) = 0$ $\Rightarrow \left(m+n\right)\left(mn\right)= 0; \Rightarrow \left(m+n\right)=0$ [Since, $m \ne0 n$ and $\ne0 ]$