1. Tardigrade
  2. Question
  3. Mathematics
  4. Given that tan θ = m ≠ 0, tan 2 θ = n ≠ 0 and tan θ + tan 2 θ = tan 3 θ, then which one of the following is correct ?

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. 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 ?

Trigonometric Functions

Solution:

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 ] $