The maximum value of cos -1 cot -2 - cot -n - 1 is

From the given relations we have ∏_i=1^n(cos α_i)=∏_i=1^n(sin α_1) ⇒∏_i=1^n(cos α_i)=∏_i=1^n(cos α_i sin α_1)=∏_i=1^n ( sin 2α_i/2 ) Since 0≤α_1 ≤1/2" and max ...

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Standard XII

Mathematics

Trigonometric Series When Angles are in A.P. or G.P.

The maximum v...

Question

The maximum value of (cos $α_{1}$ ) (cot $α_{2}$ ) …….. (cot $α_{n}$ ) = 1 is

$\frac{1}{2 n}$

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

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$1$

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

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Solution

The correct option is D
$\frac{1}{n^{\frac{n}{2}}}$

From the given relations we have

$\prod_{i = 1}^{n} (c o s α_{i}) = \prod_{i = 1}^{n} (s i n α_{1}) \Rightarrow \prod_{i = 1}^{n} (c o s α_{i}) = \prod_{i = 1}^{n} (c o s α_{i} s i n α_{1}) = \prod_{i = 1}^{n} (\frac{s i n 2 α_{i}}{2})$

Since $0 \leq α_{1} \leq \frac{1}{2 "}$ and max. value of sin $2 α$ , is 1 for all i. $\prod_{i = 1}^{n} (c o s α_{i}) = \frac{1}{2^{\frac{n}{2}}}$

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The maximum value of (cos α1) (cot α2) …….. (cot αn) = 1 is

The correct option is D 1nn2 From the given relations we have ∏ni=1(cos αi)=∏ni=1(sin α1)⇒∏ni=1(cos αi)=∏ni=1(cos αi sin α1)=∏ni=1(sin 2αi2) Since 0≤α1≤12" and max. value of sin 2α, is 1 for all i. ∏ni=1(cos αi)=12n2

The maximum value of (cos $α_{1}$ ) (cot $α_{2}$ ) …….. (cot $α_{n}$ ) = 1 is