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

$\cos \frac{\pi}{2^2} \cdot \cos \frac{\pi}{2^3} \cdot \cos \frac{\pi}{2^4} \cdots \cos \frac{\pi}{2^{10}}=$
  1. $\frac{\sin \left(\frac{\pi}{2^{10}}\right)}{512}$
  2. $\frac{\operatorname{cosec}\left(\frac{\pi}{2^{10}}\right)}{512}$
  3. $\frac{\sin \left(\frac{\pi}{2^{10}}\right)}{1024}$
  4. $\frac{\operatorname{cosec}\left(\frac{\pi}{2^{10}}\right)}{1024}$

Solution

$ \begin{aligned} & \text { (b) } \cos \frac{\pi}{2^2} \cdot \cos \frac{\pi}{2^3} \cdot \cos \frac{\pi}{2^4} \cdot \ldots \cos \frac{\pi}{2^{10}} \\ & =\frac{1}{2 \sin \left(\frac{\pi}{2^{10}}\right)} \\ & \left.\left.=\frac{1}{2 \sin \frac{\pi}{2^{10}}}\left[\cos \frac{\pi}{2^2} \cdot \cos \frac{\pi}{2^3} \cdot \cos \frac{\pi}{2^4} \cdots \cdot \sin \left(2 \cdot \frac{\pi}{2^{10}}\right)\right] \frac{\pi}{2^{10}} \cos \frac{\pi}{2^{10}}\right)\right] \\ & =\frac{1}{2^2 \sin \frac{\pi}{2^{10}}}\left[\cos \frac{\pi}{2^2} \cdot \cos \frac{\pi}{2^3} \cdots \cos \frac{\pi}{2^8} \cdot\left(\sin 2 \cdot \frac{\pi}{2^9}\right)\right] \\ & =\frac{1}{2^2 \sin \frac{\pi}{2^{10}}} \\ & =\frac{\operatorname{cosec} \frac{\pi}{2^{10}}}{2^3}\left[\cos \frac{\pi}{2^2} \cdot \cos \frac{\pi}{2^3} \cdots \cos \frac{\pi}{2^7} \cdot \frac{2}{2} \cdot\left(\sin \frac{\pi}{2^8} \cos \frac{\pi}{2^8}\right)\right] \\ & \left.=\cos \frac{\pi}{2^7} \cdot \sin \frac{\pi}{2^7}\right] \end{aligned} $ Similarly proceeding ahead, we get $ \begin{aligned} & =\frac{\operatorname{cosec} \frac{\pi}{2^{10}}}{2^9} \sin \left(2 \cdot \frac{\pi}{2^2}\right)=\frac{\operatorname{cosec}\left(\frac{\pi}{2^{10}}\right)}{512} \times \sin \frac{\pi}{2} \\ & =\frac{\operatorname{cosec}\left(\frac{\pi}{2^{10}}\right)}{512} \end{aligned} $

Asked in: AP EAMCET 2023 (19 May Shift 1)

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