Cosine plus Cosine/Proof 1

Theorem

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

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

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

$=$

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

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

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

$=$

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

$\ds \leadsto \ \ $

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

$=$

$\ds 2 \cos A \cos B$

adding $(1)$ and $(2)$

$\ds \leadsto \ \ $

$\ds \cos \alpha + \cos \beta$

$=$

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

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

$\blacksquare$