Verify the identity $$\cos 3\alpha =4\cos^3\alpha -3\cos \alpha $$

Question

Verify the identity $$\cos 3\alpha =4\cos^3\alpha -3\cos \alpha    $$

Uniteasy Admin · Accepted Answer

$$\cos 3\alpha =\cos(\alpha +2\alpha )  $$
$$=\cos \alpha \cos 2\alpha -\sin \alpha \sin 2 \alpha   $$
$$=\cos \alpha (2\cos^2 \alpha -1)-2\sin^2\alpha \cos \alpha   
$$
$$=2\cos^3 \alpha -\cos \alpha  -2(1-\cos^2\alpha )\cos \alpha  $$
$$=2\cos^3 \alpha -\cos \alpha  -2\cos \alpha +2\cos^3 \alpha  $$
$$=4\cos^3 \alpha -3\cos \alpha   $$

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