Trigonometry - the Addition Formula of Sine

The addition and subtraction formula for sine:
Sin(A+B) = Sin(A)Cos(B)+Cos(A)Sin(B)
Sin(A-B) = Sin(A)Cos(B)-Cos(A)Sin(B)

Two different proofs of the addition formula is outlined below.

Proof 1:

Proof 2:

Construct a unit circle (ie, a circle with diameter = 1) with BC as it diameter, which also forms the diagonal of an inscribed quadrilateral ABDC, as shown below. Using the law of sine and definitions of sine and cosine, the addition formula for sine follows by a simple application of Ptolemy's theorem.

The subtraction formula is for your homework assignment.