Cosine substraction.
$$\tag{1}
\cos a - \cos b = -2 \sin \tfrac12(a + b) \sin \tfrac12(a - b)
$$
$$\tag{2}
\sin \tfrac12(a + b) \sin \tfrac12(a - b) = -\tfrac12 \cos a + \tfrac12 \cos b
$$
$$\tag{3}
\sin \alpha \sin \beta = -\tfrac12 \cos (\alpha + \beta) + \tfrac12 \cos (\alpha - \beta)
$$
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