For let and a real number k be such that Then, the value of k is equal to

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Solution

Q.
For $α, β \in (0, \frac{π}{2}),$ let $3 \sin (α + β) = 2 \sin (α - β)$ and a real number $k$ be such that $\tan α = k$ $\tan β .$ Then, the value of $k$ is equal to [2024]

1 $- \frac{2}{3}$

2 $\frac{2}{3}$

3 $5$

4 $- 5$

Ans.
(4)

We have, $3 \sin (α + β) = 2 \sin (α - β)$

$\Rightarrow \frac{\sin (α + β)}{\sin (α - β)} = \frac{2}{3} \Rightarrow \frac{\sin (α + β) + \sin (α - β)}{\sin (α + β) - \sin (α - β)} = \frac{5}{- 1}$ (Using componendo and dividendo)

$\Rightarrow \frac{2 \sin α \cos β}{2 \cos α \sin β} = - 5 \Rightarrow \frac{\tan α}{\tan β} = - 5$

$\Rightarrow \tan α = - 5 \tan β$ $∴$ $k = - 5$

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