Pythagorean Identities

The Pythagorean Identities are considered to be fundamental identities in trigonometry. They express the Pythagorean Theorem in trigonometric terms. The main Pythagorean Identity is:

Let's look at how it works. Given the unit circle, which has a radius of 1, and any point on the circle that creates the vertex of a right triangle can be represented by the coordinates (x, y)


Recall the Pythagorean Theorem is x2 + y2 = c2. Since the legs of the right triangle can be represented by sin θ and cos θ and the radius is the hypotenuse we can use the Pythagorean Theorem to derive sin2 θ + cos2 θ = 1.

From this fundamental identity we can derive equivalent identities such as:

  • sin2 θ + cos2 θ = 1

  • -cos2 θ     -cos2 θ    using the subtraction property of equality

    sin2 θ = 1 - cos2 θ

  • sin2 θ + cos2 θ = 1

  • -sin2 θ     -sin2 θ    using the subtraction property of equality

    cos2 θ = 1 - sin2 θ

  •     using the division property of equality

  • 1 + cot2 θ = csc2 θ    use substitution

    -1-1use subtraction property of equality

    cot2 θ = csc2 θ - 1

Deriving the Pythagorean identities leads to understanding the basic trigonometric proofs. Recall that when solving an equation both sides of the equation may be manipulated to find a solution. However, when proving an identity only one side of the identity may be manipulated.

Related Links:
Cofunction Identities
Sum and Difference of Angles Identities

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