You will need to watch #12 | Trigonometry: Addition of Angles - to understand how the formula written just now can be deduced.
In short, you will learn derivation of formula for about sin (A+B) and and sin (A-B) in that video.
After that, compute sin (A+B) - Sin (A-B) using the formula above: ------ (1)
Let's assume C = A+B and D = A-B, then A = (C+D)/2 & B = (C-D)/2
Finally replace all A & B in term of C & D to get the formula for Sin C - Sin D = 2 * Sin (C-D)/2 * Cos (C+D)/2 from equation (1) above
The above final formula is applied just now on the board.