\textbf{Product to sum formulas}

By summing formulas \[\sin(\alpha+\beta)\] and \[\sin(\alpha-\beta)\]:

,,\qquad 2\sin(\alpha)\cos(\beta) = \sin(\alpha + \beta) + \sin(\alpha - \beta)
,,\qquad 2\cos(\alpha)\sin(\beta) = \sin(\alpha + \beta) - \sin(\alpha - \beta)

By summing formulas \[\cos(\alpha+\beta)\] and \[\cos(\alpha-\beta)\]:

,,\qquad 2\sin(\alpha)\sin(\beta) = \cos(\alpha - \beta) - \cos(\alpha + \beta)
,,\qquad 2\cos(\alpha)\cos(\beta) = \cos(\alpha - \beta) + \cos(\alpha + \beta)
 

Tangent formula can be obtained by dividing the previous two formulas

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