## ** 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 equivalent (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

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*Mohamed A. Khamsi *

Tue Dec 3 17:39:00 MST 1996

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