Compound angle formula

A compound angle formula or addition formula is a trigonometric identity which expresses a trigonometric function of \((A+B)\) or \((A-B)\) in terms of trigonometric functions of \(A\) and \(B\).

The three basic formulae are:

\[\begin{align*} \cos(A\pm B)&=\cos A\cos B \mp \sin A\sin B\\ \sin(A\pm B)&=\sin A\cos B \pm \cos A\sin B\\ \tan(A\pm B)&=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B} \end{align*}\]

There are corresponding results for hyperbolic functions which can be obtained by applying Osborn’s Rule:

\[\begin{align*} \cosh(A\pm B)&=\cosh A\cosh B \pm \sinh A\sinh B\\ \sinh(A\pm B)&=\sinh A\cosh B \pm \cosh A\sinh B\\ \tanh(A\pm B)&=\dfrac{\tanh A\pm\tanh B}{1\pm\tanh A\tanh B} \end{align*}\]