Sum to Product: cos A − cos B

This identity rewrites a difference of two cosines as a product of two sines, which is often easier to simplify or factor. It comes from applying the cosine angle-sum and angle-difference formulas to A = ((A+B)/2)+((A-B)/2) and B = ((A+B)/2)-((A-B)/2), then subtracting the results so the cosine terms cancel and the sine terms combine. The minus sign and the two sine factors reflect that the size of the difference depends on both the average angle and the separation between the angles.

Use this when you need to factor expressions like cos A − cos B, especially in trigonometric proofs, equation solving, or simplifying oscillation/wave expressions. It is also useful when A and B are close together, because the formula makes the small-angle behavior of the difference more transparent.

• Forgetting the negative sign in front of the 2
• Mixing up the average angle (A+B)/2 with the half-difference (A−B)/2
• Using cosine instead of sine in the product after rewriting