`sin^8theta +cos^8theta`
`=(sin^4theta+cos^4theta)^2 - 2sin^4thetacos^4theta`
`=((sin^2theta+cos^2theta)^2 - 2sin^2thetacos^2theta)^2- 2sin^4thetacos^4theta`
`=(1 - 2sin^2thetacos^2theta)^2- 2sin^4thetacos^4theta`
`=1+4sin^4thetacos^4theta - 4sin^2thetacos^2theta- 2sin^4thetacos^4theta`
`=2sin^4thetacos^4theta - 4sin^2thetacos^2theta +1`
`=2sin^2thetacos^2theta(sin^2thetacos^2theta -2) +1.`
