Question

Prove that: `sin^4(pi/8)+sin^4((3pi)/8)+sin^4((5pi)/8)+sin^4((7pi)/8)=3/2`

Answer 1

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