Max value of the product

Maximum value of

\cos(\alpha_1)\cos(\alpha_2)...\cos(\alpha_n) under the restrictions 0\leq \alpha_1, \alpha_2, .....\alpha_n \leq \pi/2 and cot(\alpha_1)cot(\alpha_2) .....cot(\alpha_n) = 1
is?

3 Answers

1
Optimus Prime ·

i will call alpha as A

cotA1.cotA2.........cotAn=1

cosA1.cosA2,......cosAn=sinA1..........sinAn=X

X2=(cosA1.cosA2....cosAn)(sinA1.sinA2....sinAn)

= 1/2n(sin2A1)(sin2A2)....(sin2An)
X2≤1/2n since sin2A1≤1

X≤1/2n/2

hence maximum value of given expression is 1/2n/2

62
Lokesh Verma ·

can you prove that the equality holds?

1
Optimus Prime ·

is my answer wrong?

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