Question

If the value of
$cos \,\frac{\pi}{2^{2}}. cos \frac{\pi}{2^{3}} \cdot\cdots\cdot cos\,\frac{\pi}{2^{10}}\cdot sin\,\frac{\pi}{2^{10}}=\frac{1}{2^{k}}$, then $k$ is equal to

Answer 1

$A=cos \frac{\pi}{2^{2}}\cdot cos \frac{\pi}{2^{3}}\ldots cos \frac{\pi}{2^{10}}\cdot sin \frac{\pi}{10}$
$=\frac{1}{2}\left(cos \frac{\pi}{2^{2}}\cdot cos \frac{\pi}{2^{3}}\ldots cos \frac{\pi}{2^{9}} sin \frac{\pi}{2^{9}}\right)$
$=\frac{1}{2^{8}}\left(cos \frac{\pi}{2^{2}}\cdot sin \frac{\pi}{2^{2}}\right)=\frac{1}{2^{9}} sin \frac{\pi}{2}$
$=\frac{1}{512}=\frac{1}{2^{9}}$