The value of cos frac{pi}{2^2} cdot cos frac{ | vedclass

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Question

(A) Let $P = \cos \frac{\pi}{2^2} \cdot \cos \frac{\pi}{2^3} \cdot \dots \cdot \cos \frac{\pi}{2^{10}}$.Using the identity $\cos 	heta \cdot \cos 2

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$\frac{1}{512}$

Vedclass · Answer

(A) Let $P = \cos \frac{\pi}{2^2} \cdot \cos \frac{\pi}{2^3} \cdot \dots \cdot \cos \frac{\pi}{2^{10}}$.Using the identity $\cos 	heta \cdot \cos 2	heta \cdot \cos 4	heta \dots \cos 2^{n-1}	heta = \frac{\sin(2^n 	heta)}{2^n \sin 	heta}$,we set $	heta = \frac{\pi}{2^{10}}$ and $n = 9$.Then $P = \frac{\sin(2^9 \cdot \frac{\pi}{2^{10}})}{2^9 \sin(\frac{\pi}{2^{10}})} = \frac{\sin(\frac{\pi}{2})}{512 \sin(\frac{\pi}{2^{10}})} = \frac{1}{512 \sin(\frac{\pi}{2^{10}})}$.The expression given is $P \cdot \sin \frac{\pi}{2^{10}} = \frac{1}{512 \sin(\frac{\pi}{2^{10}})} \cdot \sin \frac{\pi}{2^{10}} = \frac{1}{512}$.

The value of $\cos \frac{\pi}{2^2} \cdot \cos \frac{\pi}{2^3} \cdot \dots \cdot \cos \frac{\pi}{2^{10}} \cdot \sin \frac{\pi}{2^{10}}$ is

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