Product-to-sum: 2sinAcosB = sin(A+B) + sin(A-B). Sum-to-product: sinC + sinD = 2sin$\frac{(C+D}{2}$)cos$\frac{(C-D}{2}$). These are inverses of each other. Sum-to-product is crucial for solving equations like sinx + sin3x = 0, where factoring reveals the solutions.