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prove that cos 2pi / 15 cos 4pi / 15 cos 8pi / 15 cos 14pi / 15 = 1 / 16

$L.H.S = cos \frac{2 π}{15} \cos \frac{4 π}{15} \cos \frac{8 π}{15} \cos \frac{14 π}{15} = \cos \frac{2 π}{15} \cos \frac{4 π}{15} \cos \frac{8 π}{15} (π - \frac{π}{15}) = - \cos \frac{π}{15} \cos \frac{2 π}{15} \cos \frac{4 π}{15} \cos \frac{8 π}{15} Let A = \frac{π}{15} \Rightarrow {L.H.S = – cos A cos 2A cos 2}^{2} {A cos 2}^{3} A = - \frac{\sin 2^{4} A}{2^{4} \sin A} = - \frac{\sin 16 A}{16 \sin A} = - \frac{\sin (15 A + A)}{16 \sin A} = - \frac{\sin (π + A)}{16 \sin A} = \frac{\sin A}{16 \sin A} = \frac{1}{16} = R.H.S$

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