If 5 =3 ( +2 ) =0, then ( + ) is equal to: - Vidyadip

MCQ

If $5\sin\alpha=3\sin(\alpha+2\beta)\not=0,$ then $\tan(\alpha+\beta)$ is equal to:

A
$2\tan\beta$
B
$3\tan\beta$
✓
$4\tan\beta$
D
$6\tan\beta$

✓

Answer

Correct option: C.

$4\tan\beta$

We have,
$5\sin\alpha=3\sin(\alpha+2\beta)$
$\Rightarrow\frac{5}{3}=\frac{\sin(\alpha+2\beta)}{\sin\alpha}$
$\Rightarrow\frac{5-3}{5+3}=\frac{\sin(\alpha+2\beta)-\sin\alpha}{\sin(\alpha+2\beta)+\sin\alpha} ($using componendo and dividendo$)$
$\Rightarrow\frac{2}{8}=\frac{\sin(\alpha+2\beta)-\sin\alpha}{\sin(\alpha+2\beta)+\sin\alpha}$
$\Rightarrow\frac{1}{4}=\frac{2\cos\frac{\alpha+2\beta+\alpha}{2}\sin\frac{\alpha+2\beta-\alpha}{2}}{2\sin\frac{\alpha+2\beta+\alpha}{2}\cos\frac{\alpha+2\beta-\alpha}{2}}$
$\Rightarrow\frac{1}{4}\frac{\cos(\alpha+\beta)\sin\beta}{\sin(\alpha+\beta)\cos\beta}$
$\Rightarrow\frac{1}{4}=\cot(\alpha+\beta)\tan\beta$
$\Rightarrow\frac{1}{4}=\frac{1}{\tan(\alpha+\beta)}\tan\beta$
$\therefore\tan(\alpha+\beta)=4\tan\beta$

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