∵sinα+cosα=1
∴(sinα+cosα)^2=1,
1+2sinαcosα=1,
sinαcosα=0
sin^2 αcos^2 α=0
sin^6 α+cos^6 α
=(sin^2 α+cos^2 α)^3-3sin^2 αcos^2 α(sin^2 α+cos^2 α)
=1-3sin^2 αcos^2 α
=1.