The addition formula of a real or complex function shows how the value of the function on a sum-formed variable can be expressed with the values of this function and perhaps of another function on the addends.

Examples

1. Addition formula of an additive function $f$ ,
   $f(x\!+\!y) = f(x)+f(y)$
2. Addition formula of the natural power function, i.e. the binomial theorem,
   $(x\!+\!y)^n = \sum_{j = 0}^n {n\choose j} x^{n-j}y^j\qquad(n = 0,\,1,\,2,\,\ldots)$
3. Addition formula of the exponential function,
   $e^{x+y} = e^xe^y$
4. Addition formulae of the trigonometric functions, e.g.
   $\cos(x\!+\!y) = \cos{x}\cos{y}-\sin{x}\sin{y},\footnote{ The addition formula of cosine is sometimes called ``the mother of all formulae''.}\,\,\,\, \tan(x\!+\!y) = \frac{\tan{x}+\tan{y}}{1-\tan{x}\tan{y}}$
5. Addition formulae of the hyperbolic functions, e.g.
   $\sinh(x\!+\!y) = \sinh{x}\cosh{y}+\cosh{x}\sinh{y}$
6. Addition formula of the Bessel function,
   $J_n(x\!+\!y) = \sum_{\nu=-\infty}^{\infty}J_\nu(x)J_{n-\nu}(y) \qquad(n = 0,\,\pm1,\,\pm2,\,\ldots)$

The five first of those are instances of algebraic addition formulae; e.g. $\cosh{x}$ and $\sinh{x}$ are tied together by the algebraic connection $\cosh^2{x}-\sinh^2{x} = 1$ .

One may also speak of the subtraction formulae of functions -- one example would be $e^{x-y} = \frac{e^x}{e^y}$ .