Trigonometry/For Enthusiasts/Transformation of products into sums

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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[edit]

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:

  1. The product of a sine and cosine as the sum or difference of two sines;
  2. The product of two cosines as the sum of two cosines;
  3. The product of two sines as the difference of two sines.

Sum into product[edit]

Let C = A+B and D = A-B. 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.