cos frac{pi}{2^2} cdot cos frac{pi}{2^3} cdot | vedclass

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(B) We use the formula $\prod_{k=2}^{n} \cos \frac{\pi}{2^k} = \frac{\sin \frac{\pi}{2}}{2^{n-1} \sin \frac{\pi}{2^n}}$. Here,$n = 10$. The product is

Vedclass · Accepted Answer

$\frac{\operatorname{cosec}\left(\frac{\pi}{2^{10}}\right)}{512}$

Vedclass · Answer

(B) We use the formula $\prod_{k=2}^{n} \cos \frac{\pi}{2^k} = \frac{\sin \frac{\pi}{2}}{2^{n-1} \sin \frac{\pi}{2^n}}$. Here,$n = 10$. The product is $P = \cos \frac{\pi}{2^2} \cdot \cos \frac{\pi}{2^3} \cdots \cos \frac{\pi}{2^{10}}$. Using the identity $\cos 	heta \cos 2	heta \cos 4	heta \cdots \cos 2^{n-1}	heta = \frac{\sin 2^n 	heta}{2^n \sin 	heta}$,we set $	heta = \frac{\pi}{2^{10}}$. Then the product is $\frac{\sin(2^9 \cdot \frac{\pi}{2^{10}})}{2^9 \sin(\frac{\pi}{2^{10}})} = \frac{\sin(\frac{\pi}{2})}{512 \sin(\frac{\pi}{2^{10}})}$. Since $\sin \frac{\pi}{2} = 1$,we get $P = \frac{1}{512 \sin(\frac{\pi}{2^{10}})} = \frac{\operatorname{cosec}(\frac{\pi}{2^{10}})}{512}$.

$\cos \frac{\pi}{2^2} \cdot \cos \frac{\pi}{2^3} \cdot \cos \frac{\pi}{2^4} \cdots \cos \frac{\pi}{2^{10}} = $

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