Question

$\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\cos \frac{14\pi }{15}=\frac{1}{4a}$
Find a

Answer 1

We have
L.H.S.$=\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\cos (\pi -\frac{\pi }{15})$
$=\left(\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\right)(-\cos \frac{\pi }{15})$
$=-\cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}$
$=\cos A\cos 2A\cos 2^{2}A\cos 2^{3}A$, where $A=\pi /15$
$=-\left[\frac{\sin 2^{4}A}{2^4\sin A}\right]=-\frac{\sin 16A}{2^4\sin A}$
$=-\frac{\sin (15A+A)}{16\sin A}=\frac{-\sin (\pi +A)}{16\sin A}$
$=\frac{\sin A}{16\sin A}=\frac{1}{16}=$ R.H.S.
Ans: 4