Prove the half-angle formula for sine.

The half-angle formula for sine is sin(x/2) = ±√[(1-cos(x))/2].

To prove the half-angle formula for sine, we start with the double-angle formula for sine:

sin(2x) = 2sin(x)cos(x)

Rearranging this formula, we get:

sin(x)cos(x) = (1/2)sin(2x)

Now, we substitute x/2 for x in the above formula:

sin(x/2)cos(x/2) = (1/2)sin(x)

We can rewrite sin(x/2)cos(x/2) as (1/2)sin(x) / cos(x/2), and use the identity cos(x/2) = ±√[(1+cos(x))/2]:

sin(x/2) = ±√[(1-cos(x))/2]

This is the half-angle formula for sine. The ± sign depends on the quadrant in which x/2 lies. If x/2 is in the first or second quadrant, then sin(x/2) is positive. If x/2 is in the third or fourth quadrant, then sin(x/2) is negative.