Question

Prove that: `sin(pi/14)sin((3pi)/14)sin((5pi)/14)sin((7pi)/14)sin((9pi)/14)sin((11pi)/14)sin((13pi)/14)=1/(64)`

Answer 1

`L.H.S. = sin(pi/14)sin((3pi)/14)sin((5pi)/14)sin((7pi)/14)sin((9pi)/14)sin((11pi)/14)sin((13pi)/14)`
`=sin(pi/14)sin((3pi)/14)sin((5pi)/14)sin((7pi)/14)sin(pi-(5pi)/14)sin(pi-(3pi)/14)sin(pi-pi/14)`
`=sin(pi/14)sin((3pi)/14)sin((5pi)/14)sin((7pi)/14)sin((5pi)/14)sin((3pi)/14)sin(pi/14)`
`=(sin(pi/14)sin((3pi)/14)sin((5pi)/14))^2sin((7pi)/14)`
`=(sin(pi/14)sin((3pi)/14)sin((5pi)/14))^2sin(pi/2)`
`=(sin(pi/14)sin((3pi)/14)sin((5pi)/14))^2*1`
`=(cos(pi/2-pi/14)cos(pi/2-(3pi)/14)cos(pi/2-(5pi)/14))^2`
`=(cos((3pi)/7)cos((2pi)/7)cos((pi)/7))^2`
We know, `(cos((3pi)/7)cos((2pi)/7)cos((pi)/7)) = 1/8`
`:. (cos((3pi)/7)cos((2pi)/7)cos((pi)/7))^2 = (1/8)^2 = 1/64 = R.H.S.`