The maximum value of \(\left(\cos \alpha_1\right) \cdot\left(\cos \alpha_2\right) \ldots\left(\cos \alpha_n\right)\) under the constraints \(0 \leq \alpha_1, \alpha_2, \ldots, \alpha_n \leq \frac{\pi}{2}\) and \(\left(\cot \alpha_1\right) \cdot\left(\cot \alpha_2\right) \ldots\left(\cot \alpha_n\right)=1\) is

\(\text { Here, }\left(\cot \alpha_1\right)\left(\cot \alpha_2\right) \ldots\left(\cot \alpha_n\right)=1 \)
\( \therefore \cos \alpha_1 \cdot \cos \alpha_2 \ldots \cos \alpha_n \)
\( =\sin \alpha_1 \cdot \sin \alpha_2 \ldots \sin \alpha_n \)
\( \text { Now, }\left(\cos \alpha_1 \cdot \cos \alpha_2 \ldots \cos \alpha_n\right)^2...(i)\)
\(=\left(\cos \alpha_1 \cdot \cos \alpha_2 \ldots \cos \alpha_n\right)\)
\(\left(\cos \dot{\alpha}_1 \cdot \cos \alpha_2 \ldots \cos \alpha_n\right)\)
\(=\left(\cos \alpha_1 \cdot \cos \alpha_2 \ldots \cos \alpha_n\right)\)
\(\left(\sin \alpha_1 \cdot \sin \alpha_2 \ldots \sin \alpha_n\right)\ldots[\text { From }(i)]\)
\(=\frac{1}{2^{\mathrm{n}}} \sin 2 \alpha_1 \cdot \sin 2 \alpha_2 \ldots \sin 2 \alpha_n\quad\) \(\ldots[\because \sin 2 \mathrm{~A}=2 \sin \mathrm{~A} \cos \mathrm{~A}]\)
But each of \(\sin 2 \alpha_i \leq 1\)
\(\therefore \left(\cos \alpha_1 \cdot \cos \alpha_2 \ldots \cos \alpha_n\right)^2 \leq \frac{1}{2^n}\)
But each of \(\cos \alpha_i\) is positive
\(\therefore \cos \alpha_1 \cdot \cos \alpha_2 \ldots \cos \alpha_n \leq \sqrt{\frac{1}{2^n}}=\frac{1}{2^{\frac{n}{2}}}\)