Question

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

Collected in the board: Trigonometry

Steven Zheng posted 1 month ago


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

Steven Zheng posted 1 month ago

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