Prove that `cos(2pi/15)cos(4pi/15)cos(8pi/15)cos(14pi/15)=1/16`

Explanation

` L.H.S. =cos frac{2 pi}{15} cos frac{4 pi}{15} cos frac{8 pi}{15} cos (pi-frac{pi}{15}) `
`
=(cos frac{2 pi}{15} cos frac{4 pi}{15} cos frac{8 pi}{15})(-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 2 ~A cos 2^{2} ~A cos 2^{3} ~A, where A=pi / 15`
`
=-[frac{sin 2^{4} ~A}{2^{4} sin A}]=-frac{sin 16 ~A}{2^{4} sin A} `
`
=-frac{sin (15 ~A+A)}{16 sin A}=frac{-sin (pi+A)}{16 sin A} `
`
=frac{sin A}{16 sin A}=frac{1}{16}= R.H.S. `