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

\begin{eqnarray*} \sin A + \sin B &=& 2 \sin \left( \frac{A+B}{2} \right) \cos \left (\frac{A-B}{2} \right) \\ \sin A - \sin B &=& 2 \sin \left( \frac{A-B}{2} \right) \cos \left (\frac{A+B}{2} \right) \\ \cos A + \cos B &=& 2 \cos \left( \frac{A+B}{2} \right) \cos \left (\frac{A-B}{2} \right) \\ \cos A - \cos B &=& -2 \sin \left( \frac{A+B}{2} \right) \sin \left (\frac{A-B}{2} \right) \end{eqnarray*} We prove the first two using the sine of a sum and sine of a difference formulas:

\begin{eqnarray*} \sin (X+Y) &=& \sin X \cos Y + \cos X \sin Y \\ \sin (X-Y) &=& \sin X \cos Y - \cos X \sin Y \end{eqnarray*} Adding or subtracting the two equations yields \begin{eqnarray*} \sin (X+Y) + \sin (X-Y) &=& 2 \sin X \cos Y \\ \sin (X+Y) - \sin (X-Y) &=& 2 \sin Y \cos X \end{eqnarray*} 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

\begin{eqnarray*} \sin A + \sin B &=& 2 \sin \left( \frac{A+B}{2} \right) \cos \left (\frac{A-B}{2} \right) \\ \sin A - \sin B &=& 2 \sin \left( \frac{A-B}{2} \right) \cos \left (\frac{A+B}{2} \right) \end{eqnarray*} as desired.

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

\begin{eqnarray*} \cos (X+Y) &=& \cos X \cos Y - \sin X \sin Y \\ \cos (X-Y) &=& \cos X \cos Y + \sin X \sin Y \end{eqnarray*} Adding or subtracting the two equations yields \begin{eqnarray*} \cos (X+Y) + \cos(X-Y) &=& 2 \cos X \cos Y \\ \cos (X+Y) - \cos(X-Y) &=& - 2 \sin Y \sin X \end{eqnarray*} 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

\begin{eqnarray*} \cos A + \cos B &=& 2 \cos \left( \frac{A+B}{2} \right) \cos \left (\frac{A-B}{2} \right) \\ \cos A - \cos B &=& - 2 \sin \left( \frac{A-B}{2} \right) \sin \left (\frac{A+B}{2} \right) \end{eqnarray*} as desired.

Notes

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.