Question

If $\tan \beta =\frac{2\sin \alpha \sin \gamma }{\sin (\alpha +\gamma )}$, show that $\tan \alpha ,tan\beta ,tan\gamma$ are in $H.P$.

Answer 1

$\tan \beta =\frac{2\sin \alpha \sin \gamma }{\sin \left(\alpha +\gamma \right)}$

$\Rightarrow \tan \beta =\frac{2\sin \alpha \sin \gamma }{\sin \alpha cos\gamma +cos\alpha sin\gamma }$

$\Rightarrow \tan \beta \left[\frac{\sin \alpha cos\gamma +cos\alpha sin\gamma }{2\sin \alpha \sin \gamma }\right]=1$

$\Rightarrow \frac{\tan \beta }{2}\left[\cot \alpha +\cot \gamma \right]=1$

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

$\Rightarrow \frac{1}{\tan \alpha },\frac{1}{\tan \beta },\frac{1}{\tan \gamma }$ are in A.P

$\Rightarrow \tan \alpha ,\tan \beta ,\tan \gamma$ are in H.P.