Question

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

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

Answer 1

Answer: (A)

A.P.

$tan\beta =\frac{2\sin \alpha \cdot \sin \gamma }{sin(\alpha +\gamma )}$
$\Rightarrow \frac{\sin (\alpha +\gamma )}{2\sin \alpha \cdot \sin \gamma }=cot\beta$

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

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

$\Rightarrow cot\alpha ,cot\beta$ and $cot\gamma$ are in A.P