Question

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

• A. $AP$
• B. $GP$
• C. $HP$
• D. $noneofthese$

Answer 1

Answer: (C)

$HP$

$2\sin \alpha \cos \beta \sin \gamma =\sin \beta (\sin \alpha \cos \gamma +\cos \alpha \sin \gamma )$

$\Rightarrow 2\sin \alpha \cos \beta \sin \gamma =\sin \beta \sin \alpha \cos \gamma +\sin \beta \cos \alpha \sin \gamma$

Dividing on both sides with $\left(\sin \alpha \sin \beta \sin \gamma \right)$ gives

$\frac{2}{\tan \beta }=\frac{1}{\tan \alpha }+\frac{1}{\tan \gamma }$

$\therefore \tan \alpha ,\tan \beta ,\tan \gamma$ are in $HP$