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 ?

#### Watch This

http://www.youtube.com/watch?v=-zhCYiHcVIE James Sousa: Double Angle Identities

http://www.youtube.com/watch?v=Rp61qiglwfg James Sousa: Half Angle Identities

#### Guidance

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

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

**Example A**

Rewrite as an expression without powers greater than one.

**Solution: ** While does technically solve this question, try to get the terms to sum together not multiply together.

**Example B**

Write the following expression with only and : .

**Solution:**

**Example C**

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

**Solution: **

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

**Concept Problem Revisited**

In order to fully identify you need to use the power reducing formula.

#### Vocabulary

An ** identity** is a statement proved to be true once so that it can be used as a substitution in future simplifications and proofs.

#### Guided Practice

1. Prove the power reducing identity for sine.

2. Simplify the following identity: .

3. What is the period of the following function?

**Answers:**

1. 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.

2. Here are the steps:

3. so . Since this implies the period is .

#### Practice

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 .