Question

Prove that: `(1+cospi/8)(1+cos(3pi)/8)(1+cos(5pi)/8)(1+cos(7pi)/8)=1/8`

Answer 1

`L.H.S. =(1+cos(pi/8)) (1+cos((3pi)/8))(1+cos((5pi)/8)) (1+cos((7pi)/8)) `
`=(1+cos(pi/8)) (1+cos((3pi)/8))(1+cos(pi-(3pi)/8)) (1+cos(pi -(pi)/8)) `
`=(1+cos(pi/8)) (1+cos((3pi)/8))(1-cos((3pi)/8)) (1-cos((pi)/8)) `
`=(1-cos^2(pi/8)) (1-cos^2((3pi)/8)) `
`=sin^2(pi/8)sin^2((3pi)/8)`
`=sin^2(pi/8)cos^2(pi/2-(3pi)/8)`
`=sin^2(pi/8)cos^2(pi/8)`
`=1/4(2sin(pi/8)cos(pi/8))^2`
`=1/4(sin(2*pi/8))^2`
`=1/4(sin(pi/4))^2`
`=1/4(1/sqrt2)^2`
`=1/8 = R.H.S.`