Prove the identity sinx/2=squareroot(1-cosx)/2.

The first step in the process of proving the half angle
identity is to raise to square both side, to eliminate the square
root.

[sin (x/2)]^2 = (1 - cos
x)/2

We'll multiply by 2 both
sides:

2[sin (x/2)]^2 = (1 - cos
x)

We'll write cos x = cos 2*(x/2) = 1 - 2[sin
(x/2)]^2

We'll replace cos x by the equivalent
expression:

2[sin (x/2)]^2 = 1 - {1 - 2[sin
(x/2)]^2}

2[sin (x/2)]^2 = 1 - 1 + 2[sin
(x/2)]^2

We'll eliminate like
terms:

2[sin (x/2)]^2 = 2[sin
(x/2)]^2

Since the LHS = RHS, the identity
sin (x/2) = sqrt [(1 - cos x)/2] is verified.