Question

If $tan3\theta +tan\theta =2tan2\theta$, then $\theta$ is equal to $(n\in Z)$

• A. $n\pi$
• B. $\frac{n\pi }{2}$
• C. $\frac{n\pi }{4}$
• D. $2n\pi$

Answer 1

The correct option is A $n\pi$

$\tan 3\theta -\tan 2\theta =\tan 2\theta -\tan \theta$

$\Rightarrow \frac{\sin \theta }{\cos 3\theta \cos 2\theta }=\frac{\sin \theta }{\cos 2\theta \cos \theta }$

$\Rightarrow \sin \theta \cos 2\theta \cos \theta =\sin \theta \cos 3\theta \cos \theta \Rightarrow \sin \theta \cos 2\theta (\cos \theta -\cos 3\theta )=0\Rightarrow \sin \theta \cos 2\theta \left(2\sin 2\theta \sin \theta \right)=0\Rightarrow \sin 4\theta {\sin }^2\theta =0$

$\Rightarrow \sin 4\theta =0$ or ${\sin }^2\theta =0$

$\Rightarrow \theta =\frac{n\pi }{4}$ or $\theta =n\pi$