Pythagorean Identities in Trigonometry: Definition & Examples

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

There are three trigonometry identities based off of the Pythagorean theorem. You will learn what they are in this lesson, as well as how to get from one to the other.

Definition

The Pythagorean identities in trigonometry are the three identities that come from the Pythagorean theorem. Recall that the Pythagorean theorem states that the hypotenuse squared of a right triangle is the sum of the square of each of the other two sides.

The Pythagorean theorem.
pythagorean identity

In the Pythagorean theorem, c stands for the hypotenuse, and a and b stand for the other two sides of the right triangle.

From this theorem, three identities can be determined from substituting in sine and cosine.

The First Identity

The first one looks a lot like the Pythagorean theorem.

The first Pythagorean identity.
pythagorean identity

This identity comes from looking at the unit circle. All the right triangles formed by the unit circle will have a hypotenuse of 1. With a hypotenuse of 1, and with our angle at the origin of the coordinate plane on which the unit circle is drawn, we can determine some relationships between the sides and our sine and cosine.

The unit circle triangle with a hypotenuse of 1.
pythagorean identity

Looking at this triangle with a hypotenuse of 1, I see that my ratio for sin (theta) is a/1, so I know that a=sin (theta). My cosine then is b/1, so b=cos (theta), and my c=1. Plugging these values into the Pythagorean theorem, I arrive at my first identity.

This identity is useful when you see a problem with a sine squared plus a cosine squared. You can use this identity and replace the sine squared plus the cosine squared with the 1.

Using the first Pythagorean identity.
pythagorean identity

The above example shows just one way you can use this identity. You can replace the sine squared plus cosine squared with the 1, or you can replace the 1 with sine squared plus cosine squared. It will depend on the problem and which will make the problem easier to solve.

The Second Identity

Our second identity actually comes from the first. This identity has one of the sides equaling 1.

The second Pythagorean identity.
pythagorean identity

To arrive at this identity, we divided our first identity by sine squared so as to get the one side to equal 1. We know that cosine squared divided by sine squared is cotangent squared, and we also know that 1 divided by sine squared is cosecant squared. Using these properties, we arrive at our second identity: 1 plus cotangent squared is cosecant squared.

You can use this identity the same as you would use the first one in problems.

Using the second Pythagorean identity.
pythagorean identity

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