Difference of two Angles Identities

The difference of two angles are trigonometric identities that express the cosine and sine of the difference of two angles. The two identities are given in the equations below:

The identities can be visually proven using the right triangle definitions of the sine and cosine functions as shown below.

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

Deriviations

Derive Difference of Two Angles Identities

To derive the difference of two angles identities, two right triangles are placed next to eachother so their angles sum together to be one angle and one triangle's angle is the difference of the sum and the other.

Derive Difference of Two Angles Identities (Complex Plane)

Derive the difference of two angle identities using the properties of complex numbers and Euler's formula in the complex plane.