Question

Given that $\tan \theta =m\ne 0,\tan 2\theta =n\ne 0$ 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)

Given that $\tan \theta =m$ and $\tan 2\theta =n$
We know from fundamentals that
$\Rightarrow \tan 3\theta =\frac{\tan \theta +\tan 2\theta }{1-\tan \theta \tan 2\theta }$
Since , $\tan 3\theta =\tan \theta +\tan 2\theta .....$ as given)
$\Rightarrow \tan \theta +\tan 2\theta =\frac{\tan \theta +\tan 2\theta }{1-\tan \theta \tan 2\theta }$
$\Rightarrow \left(\tan \theta +\tan 2\theta \right)\left(1-\tan \theta \tan 2\theta \right)-\left(\tan \theta +\tan 2\theta \right)=0$
$\Rightarrow \left(\tan \theta +\tan 2\theta \right)\left\{1-\tan \theta \tan 2\theta -1\right\}=0$
$\Rightarrow \left(\tan \theta +\tan 2\theta \right)\left(\tan \theta \tan 2\theta \right)=0$
$\Rightarrow \left(m+n\right)\left(mn\right)=0;\Rightarrow \left(m+n\right)=0$
[Since, $m\ne 0n$ and $\ne 0]$