Question

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

• A. A.P.
• B. G.P.
• C. H.P.
• D. none of these

Answer 1

The correct option is A A.P.

$\tan \beta =2\sin \alpha \sin \gamma cosec(\alpha +\gamma )$

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

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

Dividing both side by $\sin \alpha \sin \beta \sin \gamma$, we get

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

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

$\Rightarrow \cot \gamma +\cot \alpha =2\cot \beta$

Hence, $\cot \alpha ,\cot \beta ,\cot \gamma$ is in A.P