Derive Difference of Two Angles Identities
To derive the sum of two angles identities a triangle can be drawn with a hypotenuse of length , an adjacent side of length and an opposite side of length . Naturally, the angle of and the angle of are drawn as well. To derive expressions for the dimensions of this right triangle, another right triangle with an angle is related to the first.
Steps

Start by placing another righttriangle of hypotenuse and an angle of on top of the first righttriangle. The side lengths of this triangle is and .

Then, the dimensions of the right triangle defined by the angle and the hypotenuse of length can be calculated. For clarity, the reflected triangle’s sides are labeled.
Next we can observe that by finding the unknown lengths labeled below we have all the information needed to finish deriving the difference of two angles equations.
Here are the equations for the difference of two angles so far:

Observe that the righttriangle whose lengths we are interested in has an angle of and an hypotenuse of length .
This gives us all the lengths of all the sides in question as shown in the image below:
Finally, we can substitue the lengths into the equations from step 2 to finish deriving the two identities.