The maximum value of (cos α1).(cos α2) ...... (cos αn) under the restrictions 0 &#880

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The maximum value of (cos &#945;1).(cos &#945;2) ...... (cos &#945;n) under the restrictions 0 &#8804; &#945;1, &#945;2, ...., &#945;n &#8804; and (cot &#945;1).(cot&#945;2)....(cot&#945;n) = 1 is

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Given, cot &#945;1. cot &#945;2 .... cot &#945;n = 1 &#8658; &#8658; Let cos &#945;1.cos &#945;2.cos &#945;3 .... cos &#945;n = k .....(i) and sin &#945;1. sin &#945;2. sin &#945;3 ....sin &#945;n = k .....(ii) Again, multiply Eqs (i) and (ii), we get (cos &#945;1 . cos &#945;2. cos &#945;3 ... cos &#945;n) &#215; (sin &#945;1. sin &#945;2. sin &#945;3 ....sin &#945;n) = k2 k2 (2 sin &#945;1 cos &#945;1) (2sin &#945;2 cos &#945;2)...(2sin &#945;n cos &#945;n) &#8658; k2 = (sin 2 &#945;1) (sin 2&#945;2) ...(sin 2&#945;n) &#8804; sin 2 &#945;i &#8804; 1 for all 1 &#8804; i < n &#8658; k &#8804;

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