Question

Помогите срочно с алгеброй

Answer 1

$8Cos\alpha Cos(60^\circ+\alpha )Cos( 60^\circ-\alpha ) =\\\\=8Cos\alpha\cdot\frac{1}{2}\Big( Cos (60^\circ+\alpha-60^\circ+\alpha)+Cos(60^\circ+\alpha+60^\circ-\alpha )\Big)=\\\\=4Cos\alpha \cdot\Big(Cos2\alpha +Cos120^\circ\Big)=4Cos\alpha \Big(Cos2\alpha -\frac{1}{2}\Big) =\\\\=4Cos\alpha Cos2\alpha-2Cos\alpha=2Cos\alpha(2Cos2\alpha-1) =\\\\=2Cos\alpha\Big(2(2Cos^{2} \alpha-1)-1\Big)=2Cos\alpha\Big(4Cos^{2}\alpha-2-1\Big)=$

$=2Cos\alpha(4Cos^{2}\alpha-3)=2\underbrace{\Big(4Cos^{3}\alpha-3Cos\alpha\Big)}_{Cos3\alpha} =\boxed{2Cos3\alpha}\\\\2Cos3\alpha=2Cos3\alpha$

Тождество доказано