Prosthaphaeresis Formulas/Cosine minus Cosine

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Theorem

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

where $\sin$ denotes sine and $\cos$ denotes cosine.


Proof 1

\(\text {(1)}: \quad\) \(\ds \map \cos {A + B}\) \(=\) \(\ds \cos A \cos B - \sin A \sin B\) Cosine of Sum
\(\text {(2)}: \quad\) \(\ds \map \cos {A - B}\) \(=\) \(\ds \cos A \cos B + \sin A \sin B\) Cosine of Difference
\(\ds \leadsto \ \ \) \(\ds \map \cos {A + B} - \map \cos {A - B}\) \(=\) \(\ds -2 \sin A \sin B\) subtracting $(2)$ from $(1)$
\(\ds \leadsto \ \ \) \(\ds \cos \alpha - \cos \beta\) \(=\) \(\ds -2 \map \sin {\dfrac {\alpha + \beta} 2} \map \sin {\dfrac {\alpha - \beta} 2}\) setting $A + B = \alpha$ and $A - B = \beta$

$\blacksquare$


Proof 2

\(\ds \) \(\) \(\ds -2 \map \sin {\frac {\alpha + \beta} 2} \map \sin {\frac {\alpha - \beta} 2}\)
\(\ds \) \(=\) \(\ds -2 \frac {\map \cos {\dfrac {\alpha + \beta} 2 - \dfrac {\alpha - \beta} 2} - \map \cos {\dfrac {\alpha + \beta} 2 + \dfrac {\alpha - \beta} 2} } 2\) Werner Formula for Sine by Sine
\(\ds \) \(=\) \(\ds -\paren {\cos \frac {2 \beta} 2 - \cos \frac {2 \alpha} 2}\)
\(\ds \) \(=\) \(\ds \cos \alpha - \cos \beta\)

$\blacksquare$


Also known as

The Prosthaphaeresis Formulas are also known as:


Examples

$\cos 20 \degrees$ minus $\cos 50 \degrees$

$\cos 20 \degrees - \cos 50 \degrees = 2 \sin 50 \degrees \cos 10 \degrees$


$\cos 2 A$ minus $\cos 4 A$

$\cos 2 A - \cos 4 A = 2 \sin 3 A \sin A$


Also see


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

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