ДОКАЖИТЕ ТОЖДЕСТВО [latex]cos ^{4} a+sin ^{4} a=1-0,5sin ^{2} 2a[/latex]

[latex]\cos^4 \alpha +\sin^4 \alpha =(\cos^2 \alpha)^2 +(\sin^2 \alpha )^2=(\frac{1+\cos2 \alpha }{2})^2+(\frac{1-\cos2 \alpha }{2})^2=\\\\=\frac{(1+\cos2 \alpha )^2}{4}+\frac{(1-\cos2 \alpha)^2 }{4}=\frac{1+2\cos2 \alpha +\cos^22 \alpha +1-2\cos2 \alpha +\cos^22 \alpha }{4}=\\\\=\frac{2+2\cos^22 \alpha }{4}=\frac{2(1+\cos^22 \alpha )}{4}=\frac{1+(1-\sin^22 \alpha) }{2}=\frac{2-\sin^22 \alpha }{2}=1-0.5\sin^22 \alpha .[/latex]