Prosthaphaeresis Formulas/Sine minus Sine

Theorem

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

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

Proof 1

$\text {(1)}: \quad$

$\ds \map \sin {A + B}$

$=$

$\ds \sin A \cos B + \cos A \sin B$

Sine of Sum

$\text {(2)}: \quad$

$\ds \map \sin {A - B}$

$=$

$\ds \sin A \cos B - \cos A \sin B$

Sine of Difference

$\ds \leadsto \ \ $

$\ds \map \sin {A + B} - \map \sin {A - B}$

$=$

$\ds 2 \cos A \sin B$

subtracting $(2)$ from $(1)$

$\ds \leadsto \ \ $

$\ds \sin \alpha - \sin \beta$

$=$

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

setting $A + B = \alpha$ and $A - B = \beta$

$\blacksquare$

Proof 2

$\ds $

$\ds 2 \map \cos {\frac {\alpha + \beta} 2} \map \sin {\frac {\alpha - \beta} 2}$

$\ds $

$=$

$\ds 2 \frac {\map \sin {\dfrac {\alpha - \beta} 2 + \dfrac {\alpha + \beta} 2} + \map \sin {\dfrac {\alpha - \beta} 2 - \dfrac {\alpha + \beta} 2} } 2$

Werner Formula for Sine by Cosine

$\ds $

$=$

$\ds \sin \frac {2 \alpha} 2 - \sin \frac {2 \beta} 2$

$\ds $

$=$

$\ds \sin \alpha - \sin \beta$

$\blacksquare$

Also known as

The prosthaphaeresis formulas are also known as:

the factor formulas (or factor formulae)
Simpson's formulas (or Simpson's formulae), although this is usually used for similar results.

Examples

$\sin 60 \degrees$ minus $\sin 40 \degrees$

$\sin 60 \degrees - \sin 40 \degrees = 2 \cos 50 \degrees \sin 10 \degrees$

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

1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(29)$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.62$
1969: J.C. Anderson, D.M. Hum, B.G. Neal and J.H. Whitelaw: Data and Formulae for Engineering Students (2nd ed.) ... (previous) ... (next): $4.$ Mathematics: $4.3$ Trigonometric identities and hyperbolic functions
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): factor formulae
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): factor formulae
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): compound angle formulae (in trigonometry)
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Sums and differences
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Factor formulae