These identities are significantly more involved and less intuitive than previous identities. By practicing and working with these advanced identities, your toolbox and fluency substituting and proving on your own will increase. Each identity in this concept is named aptly. Double angles work on finding if you already know . Half angles allow you to find if you already know . Power reducing identities allow you to find if you know the sine and cosine of .

What is ?

### Double Angle, Half Angle, and Power Reducing Identities

#### Double Angle Identities

The **double angle identities** are proved by applying the sum and difference identities. They are left as review problems. These are the double angle identities.

#### Half Angle Identities

The **half angle identities** are a rewritten version of the power reducing identities. The proofs are left as review problems.

#### Power Reducing Identities

The **power reducing identities** allow you to write a trigonometric function that is squared in terms of smaller powers. The proofs are left as examples and review problems.

Power reducing identities are most useful when you are asked to rewrite expressions such as as an expression without powers greater than one. While does technically simplify this expression as necessary, you should try to get the terms to sum together not multiply together.

### Examples

#### Example 1

Earlier, you were asked to find . In order to fully identify you need to use the power reducing formula.

#### Example 2

Write the following expression with only and : .

#### Example 3

Use half angles to find an exact value of without using a calculator.

Sometimes you may be requested to get all the radicals out of the denominator.

#### Example 4

Prove the power reducing identity for sine.

Start with the double angle identity for cosine.

This expression is an equivalent expression to the double angle identity and is often considered an alternate form.

#### Example 5

Simplify the following identity: .

Here are the steps:

### Review

Prove the following identities.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Find the value of each expression using half angle identities.

11.

12.

13.

14. Show that .

15. Show that .

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.4.