sinasinb

The trigonometric identity sin(A)sin(B) can be expressed in a more computationally useful form through product-to-sum formulas. This identity relates the product of two sine functions to a difference of cosine functions. The formula is derived from the angle subtraction and addition formulas for cosine.

Specifically, the product-to-sum formula for sin(A)sin(B) is given by:

sin(A)sin(B) = 1/2 [cos(A - B) - cos(A + B)]

This formula is useful in various areas of mathematics and physics, particularly in signal processing and wave mechanics where the interaction of multiple sinusoidal waves is analyzed. By converting a product of sines into a sum or difference of cosines, it can simplify calculations, especially when dealing with integration or solving differential equations. The derivation involves manipulating the cosine sum and difference identities: cos(A + B) = cos(A)cos(B) - sin(A)sin(B) and cos(A - B) = cos(A)cos(B) + sin(A)sin(B). Subtracting the first from the second yields cos(A - B) - cos(A + B) = 2sin(A)sin(B), which rearranges to the product-to-sum formula.