Product and Sum Formulas

From the Addition Formulas, we derive the following trigonometric formulas (or identities)



Remark. It is clear that the third formula and the fourth are identical (use the property  to see it).
The above formulas are important whenever need rises to transform the product of sine and cosine into a sum. This is a very useful idea in techniques of integration.
Example. Express the product  as a sum of trigonometric functions.
Answer. We have

which gives


Note that the above formulas may be used to transform a sum into a product via the identities



Example. Express  as a product.
Answer. We have

Note that we used  .
Example. Verify the formula


Answer. We have

and

Hence

which clearly implies


Example. Find the real number x such that  and


Answer. Many ways may be used to tackle this problem. Let us use the above formulas. We have

Hence

Since  , the equation  gives  and the equation  gives  . Therefore, the solutions to the equation

are


Example. Verify the identity


Answer. We have

Using the above formulas we get

Hence

which implies

Since  , we get