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.

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.

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

Simplify .

We notice that we can factor out a :

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

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.

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.

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

In order to apply the double-angle formulas, we need to know the value of . The Pythagorean identity can be solved to give

So is either or . The information about the location of tells us it is in the 4th quadrant, where the sine is negative. Therefore, and

Of course, an alternative is to observe that we are being asked to evaluate and use the double-angle formula for tangents.