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:



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.

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