$\text{ If }0<\alpha ,\beta ,\gamma <\pi /2\text{ such that }$$\alpha +\beta +\gamma =\frac{\pi }{2}\text{ and }\cot \alpha , \cot \beta , \cot \gamma$ $\text{ are in arithmetic progression, then }$$\text{ the value of }\left(\cot \alpha ·\cot \gamma \right)\text{ is }$

$cot\alpha ,cot\beta ,cot\gamma are in A.P.\Rightarrow cot\alpha +cot\gamma =2cot\beta$

$\begin{array}{c}Also \alpha +\beta +\gamma =\frac{\pi }{2}\Rightarrow \alpha +\gamma =\frac{\pi }{2}-\beta .\\ \text{ so that }\cot (\alpha +\gamma )=\cot \left(\frac{\pi }{2}-\beta \right)\\ \Rightarrow \frac{\cot \alpha \cot \gamma -1}{\cot \alpha +\cot \gamma }=\frac{1}{\cot \beta }\\ \Rightarrow \cot \alpha \cot \gamma -1=2\Rightarrow \cot \alpha \cot \gamma =3\end{array}$