Sum to Product: sin A − sin B

This identity rewrites a difference of two sine values as a product, which is often easier to simplify or solve. It comes from adding the angle formulas for sine and cosine after rewriting A and B as their average plus or minus half-difference, so the pattern naturally separates into a cosine of the mean angle and a sine of the angular gap. The result shows that the size of the difference depends both on where the angles sit overall and on how far apart they are.

Use it when you need to simplify expressions like sin A - sin B, especially in algebraic manipulation, trig proofs, or when solving equations involving differences of sines. It is also useful in wave, signal, and vibration problems where two oscillations are compared or combined.

• Switching the sign and writing 2 sin((A+B)/2) cos((A-B)/2) instead of 2 cos((A+B)/2) sin((A-B)/2)
• Forgetting the half angles and using A+B or A-B directly
• Mixing up the order of A and B, which changes the sign of the sine half-difference term