The general solution of $\tan \theta+\tan 2 \theta=\tan 3 \theta$ is

The general solution of $\tan \theta+\tan 2 \theta=\tan 3 \theta$ is
  1. $\theta=(2 n+1) \frac{\pi}{2}, n \in Z$
  2. $\theta=n \pi, n \in Z$ or $\theta=\frac{p \pi}{3}, p \in Z$
  3. $\theta=\frac{n \pi}{5}, n \in Z$
  4. $\theta=(2 n-1) \frac{\pi}{3}, n \in Z$

Solution

$\tan 3 \theta=\tan (2 \theta+\theta)=\tan \theta+\tan 2 \theta$ $\therefore \frac{\tan 2 \theta+\tan \theta}{1-\tan 2 \theta \tan \theta}=\tan \theta+\tan 2 \theta$ $\therefore 1-\tan 2 \theta \tan \theta=1 \Rightarrow \tan 2 \theta \tan \theta=0 \Rightarrow \tan \theta=0$ $\therefore \theta=\mathrm{n} \pi, \mathrm{n} \in \mathrm{Z}$

Asked in: MHT CET 2020 (13 Oct Shift 2)

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