Question

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

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

Answer 1

The correct option is C $\frac{1}{512}$

$A=\cos \frac{\pi }{2^2}\cdot \cos \frac{\pi }{2^3}\cdot ...\cdot \cos \frac{\pi }{2^{10}}\cdot \sin \frac{\pi }{2^{10}}$
Put $\frac{\pi }{2^{10}}=x\Rightarrow \pi =2^{10}x$
$\Rightarrow A=(\cos x\cdot \cos 2x\cdot ...\cdot \cos 2^{8}x)\sin x$
As we know
$\frac{\sin 2^n\theta }{2^n\sin \theta }=\cos \theta \cdot \cos 2\theta \cdot ...\cos 2^{n-1}\theta \cdot$
Here, $n-1=8\Rightarrow n=9$
$\Rightarrow A=\frac{\sin 2^{9}x}{2^9\sin x}\cdot \sin x$
$\Rightarrow A=\frac{\sin 2^9\frac{\pi }{2^{10}}}{2^9}$
$\Rightarrow A=\frac{1}{512}$