Question

lf $\alpha +\beta +\gamma =\frac{\pi }{2}$ and $\cot \alpha ,\cot \beta ,\cot \gamma$ are in $A.P$., then the value of $\cot \alpha .\cot \gamma$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

$\cot \alpha ,\cot \beta ,\cot \gamma$ are in A.P.
$2\cot \beta =\cot \alpha +\cot \gamma =\frac{\tan \alpha +\tan \gamma }{\tan \alpha \tan \gamma }$
$2\cot \beta =\frac{\tan (\alpha +\gamma )(1-\tan \alpha \tan \gamma )}{\tan \alpha \tan \gamma }$
$2\cot \beta =\cot \beta \left(\frac{1-\tan \alpha \tan \gamma }{\tan \alpha \tan \gamma }\right)$
Let $\tan \alpha \tan \gamma =k$
$2=\frac{1-k}{k}$
$3k=1$
$\tan \alpha \tan \gamma =k=\frac{1}{3}$
$\cot \alpha \cot \gamma =3$