Question

$\cos \left(\frac{\pi }{2^2}\right)......\cos \left(\frac{\pi }{2^{12}}\right).\sin \left(\frac{\pi }{2^{12}}\right)=$

• A. $0$
• B. $\frac{1}{1024}$
• C. $\frac{1}{576}$
• D. $\frac{1}{2048}$

Answer 1

The correct option is A $0$

$\begin{array}{c}\text{Let}\\ x=\cos \left(\frac{\pi }{2^2}\right)\cos \left(\frac{\pi }{2^3}\right)\dots \cdots \cos \left(\frac{\pi }{2^{\prime 2}}\right)\cdot \sin \left(\frac{\pi }{2^{12}}\right)\\ \text{we know}2\sin A\cos A=\sin 2A\\ 2x=\cos \left(\frac{\pi }{2^2}\right)\cdot \cdots \cdot \cos \left(\frac{\pi }{2^{11}}\right)\cdot 2\cos \left(\frac{\pi }{2^{\prime 2}}\right)\sin \left(\frac{\pi }{2^{12}}\right)\\ 2x=\cos \left(\frac{\pi }{2^2}\right)\cdot \cdots \cdot \cos \left(\frac{\pi }{2^{\prime \prime }}\right)\sin \left(\frac{\pi }{2^{\prime \prime }}\right)\\ \text{similarly.}\\ 2^{10}x=2\cos \left(\frac{\pi }{2^2}\right)\sin \left(\frac{\pi }{2^2}\right)\\ 2^{10}x=\sin \left(\frac{\pi }{2}\right)\end{array}$ $\begin{array}{cc}{2}^{10}x& =1\\ x& =\frac{1}{2^{10}}\\ x& =\frac{1}{1024}\\ \text{option}B\text{is correct}& \end{array}$