Question

Find the value of $\cos \frac{\pi }{15}.\cos \frac{2\pi }{15}.\cos \frac{3\pi }{15}.\cos \frac{4\pi }{15}.\cos \frac{5\pi }{15}.\cos \frac{6\pi }{15}.\cos \frac{7\pi }{15}$.

Answer 1

$\cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{3\pi }{15}\cos \frac{4\pi }{15}\cos \frac{5\pi }{15}\cos \frac{6\pi }{15}\cos \frac{7\pi }{15}$
$=\frac{1}{2}\cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{3\pi }{15}\cos \frac{4\pi }{15}\cos \frac{5\pi }{15}\cos (\pi -\frac{8\pi }{15})$ (as cos $\frac{6\pi }{15}=\frac{1}{2}$)
$=\frac{-1}{2}\cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{3\pi }{15}\cos \frac{4\pi }{15}\cos \frac{5\pi }{15}\cos \frac{8\pi }{15}$

( as $cos(\pi -x)=-cosx$)

using trigonometric identities

$=\frac{-1}{2}\frac{\sin \left(2^4\frac{\pi }{15}\right)}{2^4\sin \frac{\pi }{15}}\times \frac{1}{2^2}\frac{\sin \left(2^2\frac{3\pi }{15}\right)}{\sin \frac{3\pi }{15}}$
$=-\frac{\sin \sin \frac{16\pi }{15}}{16\sin \frac{\pi }{15}}\times \frac{1}{2^3}\frac{\sin \left(\frac{12\pi }{15}\right)}{\sin \frac{3\pi }{15}}$
$=-\frac{1}{2^7}\left(\frac{\sin (\pi +\frac{\pi }{15})}{\sin \frac{\pi }{15}}\right)\times \left(\frac{\sin (\pi -\frac{3\pi }{15})}{\sin \frac{3\pi }{15}}\right)=\frac{1}{2^7}$