The maximum value of ( cos 1 ) . ( cos 2 ) ? ( cos n ) , under the restrictions 0 1 , 2 , … , n 2 and ( cot 1 ) . ( cot 2 ) ? ( cot n ) = 1 is [2001]

Question

The maximum value of $(\cos α_{1}) . (\cos α_{2}) \dots (\cos α_{n}),$ under the restrictions

$0 \leq α_{1}, α_{2}, \dots, α_{n} \leq \frac{π}{2} and (\cot α_{1}) . (\cot α_{2}) \dots (\cot α_{n}) = 1$ is [2001]

Smart Achievers · Accepted Answer

(1)

$Given: (\cot α_{1}) . (\cot α_{2}) \dots (\cot α_{n}) = 1$

$\Rightarrow (\cos α_{1}) (\cos α_{2}) \dots (\cos α_{n}) = (\sin α_{1}) (\sin α_{2}) \dots (\sin α_{n}) \dots (i)$

$Let y = (\cos α_{1}) (\cos α_{2}) \dots (\cos α_{n}) (to be max.)$

$\Rightarrow y^{2} = (\cos^{2} α_{1}) (\cos^{2} α_{2}) \dots (\cos^{2} α_{n})$

$= \cos α_{1} \sin α_{1} \cos α_{2} \sin α_{2} \dots \cos α_{n} \sin α_{n} (From (i))$

$= \frac{1}{2^{n}} [\sin 2 α_{1} \sin 2 α_{2} \dots \sin 2 α_{n}]$

$Now, 0 \leq α_{1}, α_{2}, \dots, α_{n} \leq \frac{π}{2}$

$\Rightarrow 0 \leq 2 α_{1}, 2 α_{2}, \dots, 2 α_{n} \leq π$

$\Rightarrow 0 \leq \sin 2 α_{1}, \sin 2 α_{2}, \dots, \sin 2 α_{n} \leq 1$

$∴$ $y^{2} \leq \frac{1}{2^{n}} \cdot 1 \Rightarrow y \leq \frac{1}{2^{n / 2}}$

$∴$ $Max. value of y i.e. (\cos α_{1}) . (\cos α_{2}) \dots (\cos α_{n}) = \frac{1}{2^{n / 2}}$

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