\textbf{Sum to product formulas}

These are obtained by setting \[x = \alpha + \beta,\enspace y = \alpha - \beta\], solving \[\alpha = \frac{x + y}{2},\enspace\beta=\frac{x - y}{2}\] then substituting this in the "product-to-sum" formulas above.
After renaming \[x, y\] back to \[\alpha,\beta\]:

,,\qquad \sin(\alpha) + \sin(\beta) = 2\sin\left(\frac{\alpha + \beta}{2}\right)\cos\left(\frac{\alpha - \beta}{2}\right)
,,\qquad \sin(\alpha) - \sin(\beta) = 2\sin\left(\frac{\alpha - \beta}{2}\right)\cos\left(\frac{\alpha + \beta}{2}\right)
,,\qquad \cos(\alpha) + \cos(\beta) = 2\cos\left(\frac{\alpha + \beta}{2}\right)\cos\left(\frac{\alpha - \beta}{2}\right)
,,\qquad \cos(\alpha) - \cos(\beta) = -2\sin\left(\frac{\alpha + \beta}{2}\right)\sin\left(\frac{\alpha - \beta}{2}\right)

Tangent formula can be obtained in the following way:

,,\qquad \tan(\alpha) + \tan(\beta) = \frac{\sin(\alpha)}{\cos(\alpha)} + \frac{\sin(\beta)}{\cos(\beta)} = \frac{\sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)}{\cos(\alpha)\cos(\beta)} = \frac{\sin(\alpha + \beta)}{\cos(\alpha)\cos(\beta)}