The sum of two angles are two trigonometric identities that express the cosine and sine of the sum of two angles in terms of their individual components. The two identities are given in the equations below:

The identities can be visualized as a “proof without words” as shown in figure below. This representation is one of multiple ways to derive the summation identities.

To derive the sum of two angles identities, two right triangles are placed next to eachother so their angles sum together, then their proportions are related together.

The sum of two angles addition formula can be derived using a quadrilateral inscribed on a circle of diameter 1.

The sum of two angles addition formula can be derived using a quadrilateral inscribed on a circle of diameter 1.

The sum of two angles identities can be derived using the properties of the complex plane and Euler's formula.

The trigonometric identites are a set of equations derived from the properties of the right triangle and the circle.