Difference to product with sum-to-product formulae

Problem

Transform into a product: $\cos(7x) - \cos(3x)$.

Solution

We apply the sum-to-product formula $\cos p - \cos q = -2\sin\!\tfrac{p+q}{2}\sin\!\tfrac{p-q}{2}$ with $p=7x$ and $q=3x$: $\cos(7x) - \cos(3x) = -2\sin\!\left(\tfrac{7x+3x}{2}\right)\sin\!\left(\tfrac{7x-3x}{2}\right) = \boxed{-2\sin(5x)\sin(2x)}.$

Links

Topics: Prostaferesi werner
Concepts: Formule di prostaferesi
Functions: Coseno
Methods: Formule prostaferesi
Skills: Scomporre · Usare formule
Exercise type: Scomposizione