Question

lf $\tan \beta =2\sin \alpha \sin \gamma \text{cosec}(\alpha +\gamma )$ , then $\cot \alpha ,\cot \beta ,\cot \gamma$ are in

• A. $A.P$
• B. $G.P$
• C. $H.P$
• D. $A.C.P$

Answer 1

The correct option is A $A.P$

we straight away get
$\frac{1}{\cot \beta }=\frac{2\sin \alpha \sin \gamma }{\sin (\alpha +\gamma )}$
so $\cot \beta =\frac{\sin \alpha \cos \gamma +\cos \alpha \sin \gamma }{2\sin \alpha \sin \gamma }$
so $2\cot \beta =\cot \alpha +\cot \gamma$
so AP is correct