The expression si$n^8$θ − co$s^8$θ is equal to:

si$n^8$θ − co$s^8$θ = (si$n^4$θ − co$s^4$θ)(si$n^4$θ + co$s^4$θ) = (si$n^2$θ − co$s^2$θ)(si$n^2$θ + co$s^2$θ)(si$n^4$θ + co$s^4$θ). The middle bracket is 1 and si$n^4$θ + co$s^4$θ = 1 − 2si$n^2$θ·co$s^2$θ.