This example demonstrates how to derive the double angle identities using the properties of complex numbers in the complex plane. The double-angle identities are shown below.

Observe that the angle (theta) forms the complex number on the unit circle given by Euler’s formula.

This complex number is visualized by the figure below:

Model the double angle (alpha) as the input to the exponential function.

Distribute the complex constant .

Apply the addition property of the exponential function to expand to the right-hand side.

Apply Euler’s formula to both sides of the equation where the exponential function appears.

Multiply the right-hand side of the equation.

Substitute for in the expression.

Equate the real parts and complex parts of the two complex numbers.

This gives us the cosine double-angle identity.

And the sine double angle identity.