[latex]sin^{6} alpha+cos^{6} alpha=(sin^2alpha)^3+(cos^2alpha)^3=[/latex] [latex](sin^2alpha+cos^2alpha)left((sin^2alpha)^2-sin^2alpha cos^2alpha+(cos^2alpha)^2 ight)=[/latex] [latex]sin^4alpha-sin^2alpha cos^2alpha+cos^4alpha=[/latex] [latex](sin^4alpha+cos^4alpha)-sin^2alpha cos^2alpha=[/latex] [latex](sin^2alpha+cos^2alpha)^2-2sin^2alpha cos^2alpha-sin^2alpha cos^2alpha=[/latex] [latex]1-3sin^2alpha cos^2alpha=[/latex] [latex]1-3(sinalpha cosalpha)^2=[/latex] [latex]1-3cdot left(frac{ sqrt{5} }{6} ight)^2=[/latex] [latex]1-3cdot frac{ 5}{36}=[/latex] [latex]1- frac{5}{12}= frac{7}{12}[/latex]
