Using Sum to Product Identities

Let \alpha = \dfrac{s+t}{2} , \beta =\dfrac{s-t}{2} , then

\cos(\alpha-\beta) = t, \cos(\alpha+\beta ) = s

Replacement in (1), (2) yields

\cos s + \cos t = 2\cos\dfrac{s +t }{2} \cos\dfrac{s -t }{2}

\cos s - \cos t = -2\sin\dfrac{s +t }{2} \sin\dfrac{s -t }{2}

So we have derived the Sum to Product Identities for Cosine Functions

\cos \alpha + \cos \beta = 2\cos\dfrac{\alpha +\beta }{2} \cos\dfrac{\alpha -\beta }{2}

\cos \alpha - \cos \beta = -2\sin\dfrac{\alpha +\beta }{2} \sin\dfrac{\alpha -\beta }{2}