Question

What is the proof of the half-angle formula?

Answer 1

Assuming that you have the Double Angle Formula for Cosine:
`cos(2theta) = 2cos^2(theta)-1`
and the Pythagorean Formula for Sines and Cosines:
`cos^2(theta)+sin^2(theta) = 1`

The Half Angle Formula for Cosine
follows directly from the Double Angle Formula for Cosine:
`cos^2(theta/2) = (1+cos(theta))/2`

The Half Angle Formula for Sine
is developed from the Half Angle Formula for Cosine (and the Pythagorean Formula)
`sin^2(theta) = 1 -cos^2(theta) " ( Pythagorean )"`

`= 1 - (1+cos(theta))/2 " (Half Angle Cosine)"`

`= (1-cos(theta))/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)