Product-to-sum formulas convert products into sums: 2sinAcosB = sin(A+B) + sin(A-B), 2cosAsinB = sin(A+B) - sin(A-B), 2cosAcosB = cos(A+B) + cos(A-B), 2sinAsinB = cos(A-B) - cos(A+B). Sum-to-product formulas are their inverses: sinC + sinD = 2sin$\frac{(C+D}{2}$)cos$\frac{(C-D}{2}$), sinC - sinD = 2cos$\frac{(C+D}{2}$)sin$\frac{(C-D}{2}$), cosC + cosD = 2cos$\frac{(C+D}{2}$)cos$\frac{(C-D}{2}$), cosC - cosD = -2sin$\frac{(C+D}{2}$)sin$\frac{(C-D}{2}$). These are essential for: solving equations like sinx + sin3x = 0 (factor into product form), simplifying series of trig terms, and proving identities. The sum-to-product direction is more frequently used in JEE problems.