This example derives the sum of two angles identities using the properties of the exponential function with complex numbers as input. This example uses Euler’s Formula, shown below, which returns a complex number on the unit circle in the complex plane in terms of sine and cosine.

Given the sum of two angles (alpha) and (beta), along with the complex constant as input, the exponential function returns the corresponding complex number on the unit circle.

Distribute the complex constant for both angles.

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

Apply Euler’s formula to both sides of the equation.

Multiply the right-hand side of the equation.

Substitute for on the right-hand side.

Equate the real part and complex parts of the right and left sides together.

Divide both sides of the second expression by to get the sum of two angles identities.