’Prosthaphaeresis’ comes from the Greek: “prosthesi” = addition + “afairo” = subtraction.

The Prosthaphaeresis formula $\cos x\cos y=\frac{\cos(x+y)+\cos(x-y)}{2}$ was used by scientists to transform multiplication into addition. For example, to calculate the product $ab$, where $0<a,b<1$ (for $a$ and $b$ outside of this range, it is a simple matter to multiply or divide by a factor of 10 and divide or multiply this back in later), one would let $\cos x=a$ and $\cos y=b$. Using a table of cosines, one could then find an approximate value for $x$ and $y$, then find $x+y$ and $x-y$, and look up the cosines of the resulting two quantities (that is, $\cos(x+y)$ and $\cos(x-y)$). The average of these numbers is the desired product $ab$. This technique was used by Tycho Brahe to perform astronomical calculations.