The value of cos(π)/(2^2)·cos(π)/(2^3)·s·scosdfracπ2^10·sindfracπ2^10 is

The value of cos(π)/(2^2)·cos(π)/(2^3)·s·scosdfracπ2^10·sin… — JEE Main 2019 Mathematics PYQ

JEE Main 2019 · 10 Jan · Shift II

The value of

\cos\dfrac{\pi}{2^2}\cdot\cos\dfrac{\pi}{2^3}\cdots\cdots\cos\dfrac{\pi}{2^{10}}\cdot\sin\dfrac{\pi}{2^{10}}

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

Show answer and solution

Correct option: C

Solution

Use the identity sin(2θ) = 2·sin(θ)·cos(θ) repeatedly.

Start from the right: sin(π/2¹⁰)·cos(π/2¹⁰) = (1/2)·sin(π/2⁹).

Multiply by cos(π/2⁹): (1/2)·sin(π/2⁹)·cos(π/2⁹) = (1/4)·sin(π/2⁸).

Each multiplication by the next cosine halves and doubles the angle. After applying this 9 times (from cos(π/2²) to cos(π/2¹⁰)), we get (1/2⁹)·sin(π/2) = (1/512)·1 = 1/512.

[JEE Main 2019 · 10 Jan · Shift II]

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