Maximum value of $\prod_{r=1}^n\cos a_r$ if $0\le a_1,\ldots,a_n\le\frac{\pi}{2}$ and $\prod_{r=1}^n\cot a_r = 1$

Question

What is the maximum value of $\prod^n_{r=1 }\cos a_r$ under the restriction $0\le a_1,a_2,\ldots,a_n \le\frac{\pi}{2}$ and $\prod^n_{r=1}\cot a_r = 1$?

This question I could not actually think of any way on how to solve it because I will not think of any function on which I could apply Jensen's Theorem or any AM -GM inequality. Or any function whose derivative I could equate to zero to get the desired result. Please help , how to solve these type of questions.

Answer 1

By AM-GM $$\prod_{r=1}^n\frac{1}{\cos^2a_r}=\prod_{r=1}^n(1+\tan^2a_r)\geq2^n\prod_{r=1}^n\tan a_r=2^n.$$ Can you end it now?