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If $\tan \alpha=2 \sin \beta \sin \gamma \operatorname{cosec}(\beta+\gamma)$, then
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$\tan \gamma, \tan \alpha, \tan \beta$ are in harmonic progression
We have, $\tan \alpha=2 \sin \beta \sin \gamma \operatorname{cosec}(\beta+\gamma)$
$\begin{array}{ll}\Rightarrow & \tan \alpha=\frac{2 \sin \beta \sin \gamma}{\sin (\beta+\gamma)} \\ \Rightarrow & \tan \alpha=\frac{2 \sin \beta \sin \gamma}{\sin \beta \cos \gamma+\cos \beta \sin \gamma} \\ \Rightarrow & \tan \alpha=\frac{2}{\cot \gamma+\cot \beta} \\ \Rightarrow & \tan \alpha=\frac{2 \tan \beta \tan \gamma}{\tan \beta+\tan \gamma}\end{array}$
$\therefore \tan \gamma, \tan \alpha$ and $\tan \beta$ are in HP.
$\begin{array}{ll}\Rightarrow & \tan \alpha=\frac{2 \sin \beta \sin \gamma}{\sin (\beta+\gamma)} \\ \Rightarrow & \tan \alpha=\frac{2 \sin \beta \sin \gamma}{\sin \beta \cos \gamma+\cos \beta \sin \gamma} \\ \Rightarrow & \tan \alpha=\frac{2}{\cot \gamma+\cot \beta} \\ \Rightarrow & \tan \alpha=\frac{2 \tan \beta \tan \gamma}{\tan \beta+\tan \gamma}\end{array}$
$\therefore \tan \gamma, \tan \alpha$ and $\tan \beta$ are in HP.
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