Question

If $\sin B=3\sin (2A+B)$, prove that $2\tan A+\tan (A+B)=0$.

Answer 1

$\sin B=3\sin (2A+B)$
$\Rightarrow \sin (A+B-A)=3\sin (A+A+B)$
$\Rightarrow \sin (A+B)\cos A-\cos (A+B)\sin \left(B\right)=3[\sin (A+B)\cos A+\cos (A+B\left)\sin \left(B\right)\right]$

divide both side by $\cos (A+B).\cos A$

$\Rightarrow \tan (A+B)-\tan \left(A\right)=3\tan (A+B)+3\tan \left(A\right)$
$\Rightarrow 4\tan \left(A\right)+2\tan (A+B)=0$
$\therefore 2\tan \left(A\right)+\tan (A+B)=0$