Multiple angle formulas

Definition

An identity that expresses the expansion of trigonometric functions with multiple angles in terms of same trigonometric functions of angle is called multiple angle identity or multiple angle formula.

List of Multiple angle identities

1

Double angle formulas

$(1)\,\,\,\,$ $\sin 2 \theta = 2\sin \theta \cos \theta = \dfrac{2 \tan \theta}{1+\tan^2 \theta}$

$(2)\,\,\,\,$ $\cos 2 \theta = \cos^2 \theta -\sin^2 \theta = 2\cos^2 \theta -1 = 1 -2\sin^2 \theta$

$(3)\,\,\,\,$ $\tan 2 \theta = \dfrac{2 \tan \theta}{1-\tan^2 \theta} = \dfrac{2}{\cot \theta -\tan \theta}$

$(4)\,\,\,\,$ $\cot 2 \theta = \dfrac{\cot^2 \theta -1}{2 \cot \theta} = \dfrac{\cot \theta -\tan \theta}{2}$

2

Triple angle formulae

$(1)\,\,\,\,$ $\sin 3\theta = 3\sin \theta -4\sin^3 \theta$

$(2)\,\,\,\,$ $\cos 3\theta = 4\cos^3 \theta -3\cos \theta$

$(3)\,\,\,\,$ $\tan 3\theta = \dfrac{3\tan \theta -tan^3 \theta}{1 -3tan^2 \theta}$

$(4)\,\,\,\,$ $\cot 3\theta = \dfrac{3\cot \theta -\cot^3 \theta}{1 -3\cot^2 \theta}$

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