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Sum-to-product identities

Here are the sum-to-product identities:

  • $\sin (x) + \sin (y) = 2 \sin \left(\frac{x + y}{2}\right) \cos \left(\frac{x - y}{2}\right)$
  • $\sin (x) - \sin (y) = 2 \sin \left(\frac{x - y}{2}\right) \cos \left(\frac{x + y}{2}\right)$
  • $\cos (x) + \cos (y) = 2 \cos \left(\frac{x + y}{2}\right) \cos \left(\frac{x - y}{2}\right)$
  • $\cos (x) - \cos (y) = -2 \sin \left(\frac{x + y}{2}\right) \sin \left(\frac{x - y}{2}\right)$

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