# 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.

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 formulas

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

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

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

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

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