Sum of two Angles Identities

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 1. This representation is one of multiple ways to derive the summation identities.

This figure illustrates the "proof without words" of the sum of two angles identities
Figure 1: Sum of Two Angles Identities

Derivations

Derive Sum of Two Angles Identities
Derive Sum of Two Angles Identities | Example

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.

Derive Sum of Two Angles (Ptolemy's Theorem)
Derive Sum of Two Angles (Ptolemy's Theorem) | Example

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

Derive Sum of Two Angles (Unit Circle)
Derive Sum of Two Angles (Unit Circle) | Example

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