# What is the proof of the half-angle formula?

Apr 25, 2015

Assuming that you have the Double Angle Formula for Cosine:
$\cos \left(2 \theta\right) = 2 {\cos}^{2} \left(\theta\right) - 1$
and the Pythagorean Formula for Sines and Cosines:
${\cos}^{2} \left(\theta\right) + {\sin}^{2} \left(\theta\right) = 1$

The Half Angle Formula for Cosine
follows directly from the Double Angle Formula for Cosine:
${\cos}^{2} \left(\frac{\theta}{2}\right) = \frac{1 + \cos \left(\theta\right)}{2}$

The Half Angle Formula for Sine
is developed from the Half Angle Formula for Cosine (and the Pythagorean Formula)
${\sin}^{2} \left(\theta\right) = 1 - {\cos}^{2} \left(\theta\right) \text{ ( Pythagorean )}$

$= 1 - \frac{1 + \cos \left(\theta\right)}{2} \text{ (Half Angle Cosine)}$

$= \frac{1 - \cos \left(\theta\right)}{2}$

Other half Angle Formulae can be developed from these.

(Caution: If converting these "squared" half angle functions by taking the square roots, be sure to adjust the sign for the quadrant of the angle)