The value of cos π 2 ^ 2 ⋅ cos π 2 ^ 3 ⋅ ⋅ cos π 2 ^ 10 ⋅ sin π 2 ^ 10 is:

Correct answer: B. (1)/(512) Given cos(π)/(22) · cos(π)/(23) · . . . · cos(π)/(210) · sin(π)/(210) = ( 2 ^ 9 . (π)/(210)) 2 ^ 9 . sin(π)/(210) . sin(π)/(210) (UsingcosA . 2A . cos 2 ^ 2 A . cos 2 ^ 3 A . cos 2 ^ 4 A . . . . . cos 2 ^ n - 1 A = sin 2 ^ n A 2 ^ n sinA ) = (1)/(29) = (1)/(512)

JEE Main

Exam: JEE Main
Subject: Mathematics
Chapter: Trigonometric Ratios & Identities

JEE Main · Mathematics · Trigonometric Ratios & Identities

The value of cos π 2 ^ 2 ⋅ cos π 2 ^ 3 ⋅ ⋅ cos π 2 ^ 10 ⋅ sin π 2 ^...

The value of

cos \dfrac{\pi}{{2}^{2}} ⋅ cos \dfrac{\pi}{{2}^{3}} ⋅ \ldots ⋅ cos \dfrac{\pi}{{2}^{10}} ⋅ sin \dfrac{\pi}{{2}^{10}}

is:

Options

A $\dfrac{1}{1024}$
B $\dfrac{1}{512}$
C $\dfrac{1}{2}$
D $\dfrac{1}{256}$

Correct answer

B. $\dfrac{1}{512}$

Step-by-step solution

Given $cos\dfrac{π}{22} \cdot cos\dfrac{π}{23} \cdot . . . \cdot cos\dfrac{π}{210} \cdot sin\dfrac{π}{210}$

$= \dfrac{\sin \left({2}^{9} . \dfrac{π}{210}\right)}{{2}^{9} . sin\dfrac{π}{210}} . sin\dfrac{π}{210}$

\left(UsingcosA . \cos 2A . cos{2}^{2}A . cos{2}^{3}A . cos{2}^{4}A . . . . . cos{2}^{n - 1}A = \dfrac{sin{2}^{n}A}{{2}^{n}sinA}\right) $$ = \dfrac{1}{29} $$ = \dfrac{1}{512} $$

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