In order to understand the proof below, you must have a prior knowledge about circular functions. The proof also uses symmetry which is I think quite basic for high schools students. The basic idea is that the point is symmetric with the point with respect to the axis.

The angles and are rotations with the positive x-axis as the initial side. The angle is a counterclockwise rotation and the angle is a clockwise rotation as shown by the arrows in the diagram below.

**Theorem**

In this short post, we are going to prove the following trigonometric identities:

.

**Proof**

The terminal side of and intersect the unit circle at points P and P’. The coordinates of

are

are .

Since and are symmetric with respect to the x-axis,

.

Now, we know that

.

And we are done.