Product-to-sum formula ($2\sin A\sin B=\cos (A-B)-\cos (A+B)$)

Product-to-sum formulas turn products of trigonometric functions into sums at combined arguments. The identity $2\sin A\sin B=\cos (A-B)-\cos (A+B)$ is especially useful for verifying orthogonality of sine Fourier series modes on an interval.