Double angle formula

A double angle formula is a trigonometric identity which expresses a trigonometric function of \(2\theta\) in terms of trigonometric functions of \(\theta\). They are special cases of the compound angle formulae. The main formulae are:

\[\begin{align*} \cos 2\theta &= \cos^2 \theta - \sin^2 \theta \\ &= 2 \cos^2 \theta - 1 \\ &= 1 - 2 \sin^2 \theta \\ \sin 2\theta &= 2 \sin \theta \cos \theta \\ \tan 2\theta &= \frac{2 \tan \theta}{1 - \tan^2 \theta} \end{align*}\]

There are corresponding formulae for the hyperbolic functions, which can be obtained by applying Osborn’s rule to these formulae.