# Prosthaphaeresis Formulas/Cosine plus Cosine

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## Contents

## Theorem

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

where $\cos$ denotes cosine.

## Proof

\(\displaystyle \) | \(\) | \(\displaystyle 2 \, \map \cos {\frac {\alpha + \beta} 2} \, \map \cos {\frac {\alpha - \beta} 2}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2 \frac {\map \cos {\dfrac {\alpha + \beta} 2 - \dfrac {\alpha - \beta} 2} + \map \cos {\dfrac {\alpha + \beta} 2 + \dfrac {\alpha - \beta} 2} } 2\) | Simpson's Formula for Cosine by Cosine | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \cos \frac {2 \beta} 2 + \cos \frac {2 \alpha} 2\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \cos \alpha + \cos \beta\) |

$\blacksquare$

## Linguistic Note

The word **prosthaphaeresis** or **prosthapheiresis** is a neologism coined some time in the $16$th century from the two Greek words:

**prosthesis**, meaning**addition****aphaeresis**or**apheiresis**, meaning**subtraction**.

With the advent of machines to aid the process of arithmetic, this word now has only historical significance.

Ian Stewart, in his *Taming the Infinite* from $2008$, accurately and somewhat diplomatically describes the word as "ungainly".

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.63$