Product Formulae (C)

We may also derive identities for $\sin(A)\sin(B)$, $\sin(A)\cos(B)$ and $\cos(A)\cos(B)$ from the addition and subtraction formulae. We have

$$\cos(A)\cos(B) = \frac{1}{2}\left(\cos(A-B) + \cos(A+B)\right),$$

$$\sin(A)\sin(B) = \frac{1}{2}\left(\cos(A-B) - \cos(A+B)\right),$$

and

$$\sin(A)\cos(B) = \frac{1}{2}\left(\sin(A-B) + \sin(A+B)\right).$$

Exercise:

Derive these identities. For the first and second identity, add and subtract the sum and difference formulae for cosine. For the third, add the sum and difference formulae for sine.