Using the Euler identity \[e^{i\theta} = \cos\theta + i\sin\theta\]:

,align\qquad \cos(\alpha + \beta) + i\sin(\alpha + \beta) &= e^{i(\alpha + \beta)} = e^{i\alpha}e^{i\beta} \\&= (\cos\alpha + i\sin\alpha)(\cos\beta + i\sin\beta)\\&=\cos\alpha\cos\beta - \sin\alpha\sin\beta + i(\sin\alpha\cos\beta + \cos\alpha\sin\beta)

Matching real and imaginary parts left and right, we obtain the angle sum formulas \[\cos(\alpha + \beta)\] and \[\sin(\alpha + \beta)\].
Difference formulas are obtained by substituting \[\beta\] for \[-\beta\] and remembering that:

,,\qquad\sin(-\theta) = -\cos(\theta)
,,\qquad\cos(-\theta) = +\cos(\theta)

\textbf{Angle sum and difference formulas}:

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

By dividing (1) and (3), or (2) and (4):

,,\qquad\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)}
,,\qquad \tan(\alpha - \beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}

And from these, other formulas can be derived.