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Given $$ 0 \le \alpha_1,\alpha_2, \cdots \alpha_n \le \frac{\pi}{2}$$ and $$ \cot(\alpha_1)\cot(\alpha_2)\cdots \cot(\alpha_n)=1$$ Find the Maximum Value of $$ \cos(\alpha_1)\cos(\alpha_2)\cdots \cos(\alpha_n)$$ NB: A small hint will suffice

manthanomen
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Umesh shankar
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  • I don't know whether this helps, but the condition on the cotangents implies that the product of the cosines equals the product of the sines. But the bigger the cosine, the smaller the sine. – Gerry Myerson Jun 04 '13 at 12:51
  • I think that if your question has been answered then you should accept the answer given or ask for more information if you want. – Freelancer Mar 14 '16 at 12:33

1 Answers1

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since $$\sin{a_{1}}\sin{a_{2}}\cdots\sin{a_{n}}=\cos{a_{1}}\cos{a_{2}}\cdots\cos{a_{n}}$$

so $$(\cos{a_{1}}\cos{a_{2}}\cdots\cos{a_{n}})^2=\dfrac{1}{2^n}\sin{2a_{1}}\sin{2a_{2}}\cdots\sin{2a_{n}}\le\dfrac{1}{2^n}$$

with equality if and only if $a_{1}=a_{2}=\cdots=a_{n}=\dfrac{\pi}{4}$

math110
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