# Explainer: Double-Angle and Half-Angle Identities

In this explainer, we will learn how to use the Pythagorean identity and double-angle formulas to evaluate trigonometric values.

Recall the Pythagorean identity for trigonometric functions:

Dividing through by gives the identity . Using instead gives .

It is often a good strategy to rewrite more complicated trigonometric expressions in terms of and when seeking simplifications.

### Example 1: Using Pythagorean Identities to Simplify Trigonometric Expressions

Simplify .

Using the Pythagorean identity on the numerator, we get

Since and , we get resulting in

The double-angle formulas give and in terms of and . We recall these.

### Definition: Double Angle Formulas

For any real number , the following formulas hold:

We can use these two to derive a formula for too. In this case, purely in terms of , giving us and

### Example 2: Simplifying Trigonometric Expressions Using Double-Angle Identities

Simplify .

We notice that we can factor out a :

Now, we will see examples on using the double-angle formulas in the opposite direction.

### Example 3: Simplifying Trigonometric Expressions Using Double-Angle Identities

Simplify

We use the same formula in the numerator and denominator: where we used the Pythagorean identity. In the denominator, now with the identity ,

Therefore,

A more challenging example is the following.

### Example 4: Using Trigonometric Identities to Find Exact Values of Trigonometric Expressions

Find the value of given , where .

What we must evaluate is

We are given the numerator: . It is therefore enough to know the value of for this specific .

To do this, we can square the sum of the sine and cosine, because so that, using our known sum, which means

Then,

A second example of such an evaluation follows.

### Example 5: Using Double Angle Identities to Evaluate a Trigonometric Expression

Find, without using a calculator, the value of given , where .