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  4. If 2 sin α cos β sin γ= sin β sin (α+γ) then tan α, tan β, tan γ are in

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Q. If $2 \sin \alpha \cos \beta \sin \gamma=\sin \beta \sin (\alpha+\gamma)$ then $\tan \alpha, \tan \beta, \tan \gamma$ are in

Trigonometric Functions

Solution:

$2 \sin \alpha \cos \beta \sin \gamma=\sin \beta \sin (\alpha+\gamma)$
$=\sin \beta(\sin \alpha \cos \gamma+\cos \alpha \sin \gamma)$
on dividing by $\sin \alpha \sin \beta \sin \gamma$, we get
$2 \cot \beta=\cot \gamma+\cot \alpha$
$\Rightarrow \frac{2}{\tan \beta}=\frac{1}{\tan \gamma}+\frac{1}{\tan \gamma}$
Hence, $\tan \alpha, \tan \beta, \tan \gamma$ are in $H.P$.