\textbf{Half-angle formulas}


,,\qquad \sin^2\left(\frac{\alpha}{2}\right) = \frac{1 - \cos(2\alpha)}{2}
,,\qquad \cos^2\left(\frac{\alpha}{2}\right) = \frac{1 + \cos(2\alpha)}{2}

Notice that both formulas above use the \[\cos(2\alpha)\]
Half-angle tangent formula is better given by:

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

By setting \[\beta = 0\]:

,,\qquad \tan\left(\frac{\alpha}{2}\right) = \frac{\sin(\alpha)}{\cos(\alpha) + 1}

By substituting \[\beta\] for \[-\beta\]

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