Sine and Cosine Subtraction Formulas
Now, we will try to prove the Sine and Cosine Subtraction Formulas using the Figure 2 below.

ABCD is a quadrilateral such that angle ABC and angle CDA are right angles and angle DAB is acute. X denotes the angle DAB and Y denotes the angle CAB. Thus angle CAD is X-Y.


The Sine Subtraction Formula states that

sin(x-y) =sin(x)cos(y) - cos(x)sin(y).

Let's see if we can prove this formula using the Figure 2 above!


Therefore, we have proved the Sine Subtraction Formula does, in fact work

Now, let's try proving the Cosine Subtraction Formula.

The Cosine Subtraction Formula states that

cos(x-y) =cos(x)cos(y) + sin(x)sin(y).


Now, we have proved that the Cosine Subtraction Formula does work.