Question

Prove that: `sinpi/5sin(2pi)/5sin(3pi)/5sin(4pi)/5=5/(16)`

Answer 1

`L.H.S. = sin(pi/5)sin((2pi)/5)sin((3pi)/5)sin((4pi)/5)`
`=sin(pi/5)sin((2pi)/5)sin((pi-(2pi)/5)sin(pi -pi/5)`
`=sin(pi/5)sin((2pi)/5)sin((2pi)/5)sin(pi/5)`
`=sin^2(pi/5)sin^2((2pi)/5)`
`=sin^2(36^@)sin^2(72^@)`
`=sin^2(36^@)cos^2(18^@)`
`=(sin(36^@)cos(18^@))^2`
`=((sqrt(10-2sqrt5))/4*(sqrt(10+2sqrt5))/4)^2`
`=(100-20)/(256)`
`=80/256`
`=5/16 = R.H.S.`