Producttosum

Product-to-sum refers to a set of trigonometric identities that convert products of sine and cosine into sums of trigonometric functions with the sum and difference of the original angles. These formulas are derived from the addition formulas for sine and cosine and are used to simplify integration, series, and other trigonometric expressions.

The standard identities include:

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

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

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

These identities allow any product of sine and cosine to be rewritten as a sum of functions with arguments A±B.

General use and applications: for any real angles A and B, a product such as sin A cos B or cos A cos B can be replaced by a sum of trigonometric functions with the sums and differences of the angles. This is useful in evaluating integrals, solving trigonometric equations, simplifying Fourier series, and reducing products of trigonometric terms in physics and engineering.

Derivation and relation to other identities: the product-to-sum formulas follow from the addition formulas sin(A±B) and cos(A±B). By solving those equations for products like sin A sin B, cos A cos B, and sin A cos B, one obtains the product-to-sum expressions. There is also a related set called sum-to-product identities that reverse the process.