Question

Prove:

${\sin }^8\theta -{\cos }^8\theta =({\sin }^2\theta -{\cos }^2\theta )(1-2{\sin }^2\theta {\cos }^2\theta )$

Answer 1

${sin}^8\theta -{cos}^8\theta$

$=({sin}^4\theta -{cos}^4\theta )({sin}^4\theta +{cos}^4\theta )$

$=({sin}^4\theta -{cos}^4\theta )({sin}^2\theta +{cos}^2\theta )[{({sin}^2\theta -{cos}^2\theta )}^2-2{sin}^2\theta {cos}^2\theta ]$

$=({sin}^2\theta -{cos}^2\theta )1-(-2{sin}^2\theta {cos}^2\theta )$