Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

Given $\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi }{2^2}·\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi }{2^3}·...·\mathrm{c}\mathrm{o}\mathrm{s}\frac{\pi }{2^{10}}·\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi }{2^{10}}$

$=\frac{\sin \left(2^9.\frac{\pi }{2^{10}}\right)}{2^9.\sin \frac{\pi }{2^{10}}}.\sin \frac{\pi }{2^{10}}$

$\left(\mathrm{Using} \cos A.\cos 2A.\cos 2^{2}A.\cos 2^{3}A.\cos 2^{4}A.....\cos 2^{n-1}A=\frac{\sin 2^{n}A}{2^n\sin A}\right)$

$=\frac{1}{2^9}$

$=\frac{1}{512}$