79. sin^6 alpha + cos^6 alpha = 1/8 (5 + 3 cos 4alpha);

sin^6 alpha + cos^6 alpha = (sin^2 alpha + cos^2 alpha)(sin^4 alpha - sin^2 alpha cos^2 alpha + cos^4 alpha) = 1 * ((sin^2 alpha + cos^2 alpha)^2 - 3 sin^2 alpha cos^2 alpha) = 1 - 3/4 sin^2 2alpha = 1 - 3/4 * (1 - cos 4alpha)/2 = 1 - 3/8 + 3/8 cos 4alpha = 5/8 + 3/8 cos 4alpha = 1/8 (5 + 3 cos 4alpha).