Prosthaphaeresis formulas

The Prosthaphaeresis formulas convert sums of sines or cosines to products of them:

$\\displaystyle\\sin A+\\sin B$	$\\displaystyle=$	$\\displaystyle 2\\sin\\left(\\frac{A+B}{2}\\right)\\cos\\left(\\frac{A-B}{2}\\right)$
$\\displaystyle\\sin A-\\sin B$	$\\displaystyle=$	$\\displaystyle 2\\sin\\left(\\frac{A-B}{2}\\right)\\cos\\left(\\frac{A+B}{2}\\right)$
$\\displaystyle\\cos A+\\cos B$	$\\displaystyle=$	$\\displaystyle 2\\cos\\left(\\frac{A+B}{2}\\right)\\cos\\left(\\frac{A-B}{2}\\right)$
$\\displaystyle\\cos A-\\cos B$	$\\displaystyle=$	$\\displaystyle-2\\sin\\left(\\frac{A+B}{2}\\right)\\sin\\left(\\frac{A-B}{2}\\right)$

We prove the first two using the sine of a sum and sine of a difference formulas:

	$\\displaystyle\\sin(X+Y)$	$\\displaystyle=$	$\\displaystyle\\sin X\\cos Y+\\cos X\\sin Y$
	$\\displaystyle\\sin(X-Y)$	$\\displaystyle=$	$\\displaystyle\\sin X\\cos Y-\\cos X\\sin Y$

Adding or subtracting the two equations yields

	$\\displaystyle\\sin(X+Y)+\\sin(X-Y)$	$\\displaystyle=$	$\\displaystyle 2\\sin X\\cos Y$
	$\\displaystyle\\sin(X+Y)-\\sin(X-Y)$	$\\displaystyle=$	$\\displaystyle 2\\sin Y\\cos X$

If we let $X=\\frac{A+B}{2}$ and $Y=\\frac{A-B}{2}$ , then $X+Y=\\frac{2A}{2}=A$ and $X-Y=\\frac{2B}{2}=B$ , and the last two equations become

	$\\displaystyle\\sin A+\\sin B$	$\\displaystyle=$	$\\displaystyle 2\\sin\\left(\\frac{A+B}{2}\\right)\\cos\\left(\\frac{A-B}{2}\\right)$
	$\\displaystyle\\sin A-\\sin B$	$\\displaystyle=$	$\\displaystyle 2\\sin\\left(\\frac{A-B}{2}\\right)\\cos\\left(\\frac{A+B}{2}\\right)$

as desired.

The last two can be proven similarly, this time using the cosine of a sum and cosine of a difference formulas:

	$\\displaystyle\\cos(X+Y)$	$\\displaystyle=$	$\\displaystyle\\cos X\\cos Y-\\sin X\\sin Y$
	$\\displaystyle\\cos(X-Y)$	$\\displaystyle=$	$\\displaystyle\\cos X\\cos Y+\\sin X\\sin Y$

Adding or subtracting the two equations yields

	$\\displaystyle\\cos(X+Y)+\\cos(X-Y)$	$\\displaystyle=$	$\\displaystyle 2\\cos X\\cos Y$
	$\\displaystyle\\cos(X+Y)-\\cos(X-Y)$	$\\displaystyle=$	$\\displaystyle-2\\sin Y\\sin X$

Again, if we let $X=\\frac{A+B}{2}$ and $Y=\\frac{A-B}{2}$ , then $X+Y=\\frac{2A}{2}=A$ and $X-Y=\\frac{2B}{2}=B$ , and the last two equations become

	$\\displaystyle\\cos A+\\cos B$	$\\displaystyle=$	$\\displaystyle 2\\cos\\left(\\frac{A+B}{2}\\right)\\cos\\left(\\frac{A-B}{2}\\right)$
	$\\displaystyle\\cos A-\\cos B$	$\\displaystyle=$	$\\displaystyle-2\\sin\\left(\\frac{A-B}{2}\\right)\\sin\\left(\\frac{A+B}{2}\\right)$

as desired.

Notes

’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 $a b$ , 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 $a b$ . This technique was used by Tycho Brahe to perform astronomical calculations.