Упростить выражение [latex]sin^6\alpha + cos^6 \alpha + \frac{3-3cos^2\alpha}{1+tg^2\alpha} [/latex]

[latex]1+\tan^2 \alpha=1/\cos^2 \alpha[/latex] [latex]3-3\cos^2 \alpha=3(1-\cos^2 \alpha)=3\sin^2\alpha[/latex]   [latex]\sin^6\alpha+\cos^6\alpha+\dfrac{3-3\cos^2\alpha}{1+\tan^2\alpha}=\sin^6\alpha+(1-\sin^2\alpha)^3+3\sin^2\alpha\cos^2\alpha=[/latex] [latex]=\sin^6\alpha+1-3\sin^2\alpha+3\sin^4\alpha-\sin^6\alpha+3\sin^2\alpha\cos^2\alpha=[/latex] [latex]=1-3\sin^2\alpha(1-\sin^2\alpha)+3\sin^2\alpha\cos^2\alpha=1-3\sin^2\alpha\cos^2\alpha+3\sin^2\alpha\cos^2\alpha=1[/latex]