Question

If $\sin \theta =3\sin (\theta +2\alpha ),$ then the value of $\tan (\theta +\alpha )+2\tan \alpha$ is

• A. $3$
• B. $2$
• C. $1$
• D. $0$

Answer 1

The correct option is D $0$

Given, $\sin \theta =3\sin (\theta +2\alpha )$

$\Rightarrow \sin (\theta +\alpha -\alpha )=3\sin (\theta +\alpha +\alpha )$

$\Rightarrow \sin (\theta +\alpha )\cos \alpha -\cos (\theta +\alpha )\sin \alpha =3\sin (\theta +\alpha )\cos \alpha +3\cos (\theta +\alpha )\sin \alpha .$

$\Rightarrow -2\sin (\theta +\alpha )\cos \alpha =4\cos (\theta +\alpha )\sin \alpha$

$\Rightarrow -\frac{\sin (\theta +\alpha )}{\cos (\theta +\alpha )}=\frac{2\sin \alpha }{\cos \alpha }$

$\Rightarrow -\tan (\theta +\alpha )=2\tan \alpha$

$\Rightarrow \tan (\theta +\alpha )+2\tan \alpha =0.$