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

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Domain and Range of Trigonometric Ratios

The maximum v...

Question

The maximum value of $c o s α_{1} . c o s α_{2} . . . . . . c o s α_{n},$
under the restrictions $0 \leq α_{1} α_{2}, . . . . ., α_{n} \leq \frac{π}{2} a n d c o t α_{1} . c o t α_{2} . . . . . . c o t α_{n} = 1$ is

\frac{1}{2^{\frac{n}{2}}}

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\frac{1}{2}

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\frac{1}{2 n}

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1

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Solution

The correct option is A $\frac{1}{2^{\frac{n}{2}}}$
$H e r e (c o t α_{1}) . (c o t α_{2}) . . . . . (c o t α_{n}) = 1$
$∴ c o s α_{1} . c o s α_{2} . . . c o s α_{n} = s i n α_{1} . s i n α_{2} . . . s i n α_{n}$
Now, $(c o s α_{1} . c o s α_{2} . . . c o s α_{n})^{2}$
$= (c o s α_{1} . c o s α_{2} . . . c o s α_{n}) = (c o s α_{1} . c o s α_{2} . . . c o s α_{n})$
$= (c o s α_{1} . c o s α_{2} . . . c o s α_{n}) (s i n α_{1} . s i n α_{2} . . . s i n α_{n})$
$= \frac{1}{2^{n}} s i n 2 α_{1} . s i n 2 α_{2} . . . s i n 2 α_{n} .$
But each of $s i n 2 α_{1} \leq 1$
$∴ (c o s α_{1} . c o s α_{2} . . . c o s α_{n})^{2} \leq \frac{1}{2^{n}} .$
But each of $c o s α_{1}$ is positive.
$∴ c o s α_{1} . c o s α_{2} . . . . c o s α_{n} \leq \sqrt{\frac{1}{2^{n}}} = \frac{1}{2^{\frac{n}{2}}} .$

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The maximum value of cosα1.cosα2...... cos αn, under the restrictions 0≤α1α2,.....,αn≤π2 and cotα1.cotα2......cot αn=1 is

The maximum value of $c o s α_{1} . c o s α_{2} . . . . . . c o s α_{n},$
under the restrictions $0 \leq α_{1} α_{2}, . . . . ., α_{n} \leq \frac{π}{2} a n d c o t α_{1} . c o t α_{2} . . . . . . c o t α_{n} = 1$ is