262Ans . 2-n/2
find the max value of (cos\,\, \alpha _{1}) (cos\,\, \alpha _{2}) ...... (cos\,\, \alpha _{n})
under the restrictions 0\leq \alpha _{i}\leq \Pi /2
and (cot\,\, \alpha _{1}) (cot\,\, \alpha _{2}) ...... (cot\,\, \alpha _{n}) =1
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2 Answers
1\cos^2{\alpha_1}+\cos^2{\alpha_2}+\cdots+\cos^2{\alpha_n} \geq n\left(\cos{\alpha_1}.\cos{\alpha_2}\cdots\cos{\alpha_n} \right)^{\frac{2}{n}} \\ \sin^2{\alpha_1}+\sin^2{\alpha_2}+\cdots+\sin^2{\alpha_n} \geq n\left(\sin{\alpha_1}.\sin{\alpha_2}\cdots\sin{\alpha_n} \right)^{\frac{2}{n}}\\ \texttt{Adding these two inequalities} \\ n \geq 2n\left(\cos{\alpha_1}.\cos{\alpha_2}\cdots\cos{\alpha_n } \right)^{\frac{2}{n}}\\ \left(\cos{\alpha_1}.\cos{\alpha_2}\cdots\cos{\alpha_n} \right)\leq 2^{-\frac{n}{2}}