# Simpson's Formulas

## Theorem

### Cosine by Cosine

$\cos \alpha \cos \beta = \dfrac {\map \cos {\alpha - \beta} + \map \cos {\alpha + \beta} } 2$

### Sine by Sine

$\sin \alpha \sin \beta = \dfrac {\map \cos {\alpha - \beta} - \map \cos {\alpha + \beta} } 2$

### Sine by Cosine

$\sin \alpha \cos \beta = \dfrac {\map \sin {\alpha + \beta} + \map \sin {\alpha - \beta} } 2$

### Cosine by Sine

$\cos \alpha \sin \beta = \dfrac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2$

### Hyperbolic Cosine by Hyperbolic Cosine

$\cosh x \cosh y = \dfrac {\cosh \paren {x + y} + \cosh \paren {x - y} } 2$

### Hyperbolic Sine by Hyperbolic Sine

$\sinh x \sinh y = \dfrac {\map \cosh {x + y} - \map \cosh {x - y} } 2$

### Hyperbolic Sine by Hyperbolic Cosine

$\sinh x \cosh y = \dfrac {\sinh \paren {x + y} + \sinh \paren {x - y} } 2$

## Also known as

Simpson's formulas are also known as the product formulas or product formulae.

## Source of Name

This entry was named for Thomas Simpson.