Werner formulae: from product to sum

The Werner formulae transform a product of sines and cosines into a sum (or difference). They are useful, for example, when a product has to be integrated or simplified.

Property — Werner formulae: product → sum

$\begin{aligned} \cos\alpha\cos\beta &= \tfrac{1}{2}\bigl[\cos(\alpha-\beta) + \cos(\alpha+\beta)\bigr] \\ \sin\alpha\sin\beta &= \tfrac{1}{2}\bigl[\cos(\alpha-\beta) - \cos(\alpha+\beta)\bigr] \\ \sin\alpha\cos\beta &= \tfrac{1}{2}\bigl[\sin(\alpha+\beta) + \sin(\alpha-\beta)\bigr] \end{aligned}$

Proof

For the first, we start from the addition formulae for the cosine: $\cos(\alpha-\beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta,\qquad \cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta.$ Adding term by term gives $\cos(\alpha-\beta) + \cos(\alpha+\beta) = 2\cos\alpha\cos\beta$, whence the first formula. Subtracting term by term yields the second. The third is obtained by adding the formulae for $\sin(\alpha+\beta)$ and $\sin(\alpha-\beta)$. ∎

Links

Topics: Prostaferesi werner
Concepts: Formule di addizione · Formule di werner
Functions: Coseno · Seno
Skills: Dimostrare · Usare formule