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Pythagorean Identities

The Pythagorean identities come from applying the Pythagorean theorem to a right triangle with hypotenuse 1, giving sine squared theta plus cosine squared theta equals 1. Dividing by cosine squared or sine squared gives two more forms. Learn all three and how to use them to simplify trig expressions.

Where the Pythagorean identity comes from

This relationship also underlies the unit circle, where every point is (cos θ, sin θ) and sits at distance 1 from the center.

The Pythagorean identities come directly from the Pythagorean theorem applied to a right triangle with hypotenuse 1. If one leg is sin θ and the other is cos θ, the Pythagorean theorem gives sin²θ + cos²θ = 1², or simply sin²θ + cos²θ = 1. This is true for every angle θ, not just the ones in a specific triangle.

A right triangle with hypotenuse 1 shows why sin²θ + cos²θ = 1, and the two related identities.

The other two Pythagorean identities

Dividing the main identity by cos²θ gives 1 + tan²θ = sec²θ. Dividing instead by sin²θ gives 1 + cot²θ = csc²θ. All three forms describe the same underlying relationship, just rearranged using the quotient identities.

Using the identities to simplify expressions

Because sin²θ + cos²θ always equals 1, you can substitute one side for the other whenever it simplifies an expression. For example, 1 − sin²θ simplifies directly to cos²θ, since sin²θ + cos²θ = 1 rearranges to cos²θ = 1 − sin²θ.

Worked example

Simplify sin²θ + cos²θ + tan²θ. The first two terms combine to 1 by the Pythagorean identity, leaving 1 + tan²θ — which is itself the second Pythagorean identity, so the expression simplifies all the way to sec²θ. This kind of substitution is also the starting point for the sum and difference identities.

Related lessons

Trigonometry