Product to sum with Werner

Problem

Transform into a sum: $\cos(5x)\cos(3x)$.

Solution

We apply the Werner formula $\cos\alpha\cos\beta = \tfrac{1}{2}\bigl[\cos(\alpha-\beta)+\cos(\alpha+\beta)\bigr]$ with $\alpha=5x$ and $\beta=3x$: $\cos(5x)\cos(3x) = \tfrac{1}{2}\bigl[\cos(5x-3x)+\cos(5x+3x)\bigr] = \boxed{\tfrac{1}{2}\bigl[\cos(2x)+\cos(8x)\bigr]}.$

Links

Topics: Prostaferesi werner
Concepts: Formule di werner
Functions: Coseno
Skills: Usare formule
Exercise type: Calcolo espressioni