The double angle identities express the cosine and sine of a double angle in terms of the sine and cosine of the single angle. The identities can be derived three ways: 1) By using the previously derived theorems on this page such as Pythagorean’s Identity and the Sum of Two Angles identities. 2) By using the geometry of the inscribed angle theorem and the formula for area of a triangle. 3) By using the complex plane and the properties of complex numbers.

The figure above demonstrates the inscribed angle theorem and the properties of similar triangles can be used to derive the double angle identities:

This example derives the double angle identities using algebra and the sum of two angles identities.

The double angle identities can be derived using the inscribed angle theorem on the circle of radius one.

The trigonometric double angle identites can be derived using the properties of the complex plane.