The maximum value of cos α 1 .cos α 2 ........cos α n , under the restrictions 0 ≤ α 1 , α 2 ,......... α n ≤ π 2 and cot α 1 .cot α 2 ........cot α n = 1 is Options: A . &nbsp12n/2 B . &nbsp12n C . &nbsp12n D . &nbsp1

The maximum value of cosα1.cosα2........cosαn, under the restr

Question

The maximum value of cos

α_{1}

.cos

α_{2}

........cos

α_{n}

, under the restrictions 0

\leq

α_{1}

α_{2}

,.........

α_{n}

\leq

\frac{π}{2}

and
cot

α_{1}

.cot

α_{2}

........cot

α_{n}

= 1 is

Options:

A . 12n/2

B . 12n

C . 12n

D . 1

Answer: Option A
:
A
(a) Here (cot

α_{1}

).(cot

α_{2}

)....(cot

α_{n}

)= 1

∴

cos

α_{1}

.cos

α_{2}

........cos

α_{n}

= sin

α_{1}

.sin

α_{2}

........sin

α_{n}

Now, (cos

α_{1}

.cos

α_{2}

........cos

α_{n}

)

^{2}

= (cos

α_{1}

.cos

α_{2}

........cos

α_{n}

) (cos

α_{1}

.cos

α_{2}

........cos

α_{n}

)
= (cos

α_{1}

.cos

α_{2}

........cos

α_{n}

) (sin

α_{1}

.sin

α_{2}

........sin

α_{n}

)
=

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

sin 2

α_{1}

.sin 2

α_{2}

........sin 2

α_{n}

But each of sin 2

α_{i}

\leq

1
(cos

α_{1}

.cos

α_{2}

........cos

α_{n}

)

^{2}

\leq

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

But each of cos

α_{i}

, is positive.

∴

cos

α_{1}

.cos

α_{2}

........cos

α_{n}

\leq

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

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

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