Trigonometry/For Enthusiasts/Transformation of products into sums

In this section, we shall see how to convert a product of two trigonometric functions into a sum or difference of two such functions, and vice versa. Product into sum We have already seen that and Adding these two equations and dividing both sides by 2, we get Subtracting the second from the first equation and dividing both sides by 2, we get We also know that and Adding these two equations and dividing both sides by 2, we get Subtracting the first from the second equation and dividing both sides by 2, we get Thus we can express: The product of a sine and cosine as the sum or difference of two sines; The product of two cosines as the sum of two cosines; The product of two sines as the difference of two sines. Sum into product Let and . Then Substituting into the above expressions and multiplying both sides by two in each of them, we have: Note the negative sign in the last formula. These formulae are sometimes expressed in words, e.g. cos plus cos = two cos half sum cos half diff.