Can you solve problems that involve the sum of sines or cosines? For example, consider the equation:

You could just compute each expression separately and add their values at the end. However, there is an easier way to do this. You can simplify the equation first, and then solve.

Read this Concept, and at the end of it, you'll be able to simplify this equation and transform it into a product of trig functions instead of a sum!

### Watch This

In the first portion of this video, you'll learn about the Sum to Product formulas.

James Sousa: Sum to Product and Product to Sum Identities

### Guidance

In some problems, the product of two trigonometric functions is more conveniently found by the sum of two trigonometric functions by use of identities such as this one:

This can be verified by using the sum and difference formulas:

The following variations can be derived similarly:

Here are some examples of this type of transformation from a sum of terms to a product of terms.

#### Example A

Change into a product.

**
Solution:
**
Use the formula

#### Example B

Change into a product.

**
Solution:
**
Use the formula

#### Example C

Change to a sum.

**
Solution:
**
This is the reverse of what was done in the previous two examples. Looking at the four formulas above, take the one that has sine and cosine as a product,
Therefore,
and
.

So, this translates to . A shortcut for this problem, would be to notice that the sum of and is and the difference is .

### Vocabulary

**
Sum to Product Formula:
**
A
**
sum to product formula
**
relates the sum or difference of two trigonometric functions to the product of two trigonometric functions.

### Guided Practice

1. Express the sum as a product:

2. Express the difference as a product:

3. Verify the identity (using sum-to-product formula):

**
Solutions:
**

1. Using the sum-to-product formula:

2.

Using the difference-to-product formula:

3.

Using the difference-to-product formulas:

### Concept Problem Solution

Prior to learning the sum to product formulas for sine and cosine, evaluating a sum of trig functions, such as

might have been considered difficult. But you can easily transform this equation into a product of two trig functions using:

Substituting the known quantities:

### Practice

Change each sum or difference into a product.

Change each product into a sum or difference.

- Show that .
- Let and . Show that