• # question_answer If $0<\alpha ,\,\,\beta ,\,\,\gamma <\pi /2$ such that $\alpha +\beta +\gamma =\frac{\pi }{2}$and$\cot \alpha ,\,\,\cot \beta ,\,\,\cot \gamma$ are in arithmetic progression, then the value of $\cot \alpha \cot \gamma$ is A) $1$                             B) $3$C) ${{\cot }^{2}}\beta$                     D) $\cot \alpha +\cot \gamma$

$\alpha +\beta +\gamma =\frac{\pi }{2}\Rightarrow \alpha +\gamma =\frac{\pi }{2}-\beta$ so that$\cot \left( \alpha +\gamma \right)=\cot \left( \frac{\pi }{2}-\beta \right)$ $\Rightarrow$$\frac{\cot \alpha \cot \gamma -1}{\cot \alpha +\cot \gamma }=\frac{1}{\cot \beta }$ $\Rightarrow$$\cot \alpha \cot \gamma -1=2\Rightarrow \cot \alpha \cot \gamma =3$. (since$\cot \alpha +\cot \gamma =2\cot \beta )$