Prove the half-angle formula for cosine.

The half-angle formula for cosine is cos²(x/2) = (1 + cos(x))/2.

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

cos(2x) = cos²(x) - sin²(x)

Rearranging this formula, we get:

cos²(x) = (cos(2x) + sin²(x))

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

cos²(x/2) = (cos(x) + sin²(x/2))

Using the Pythagorean identity sin²(x/2) = (1 - cos(x))/2, we get:

cos²(x/2) = (cos(x) + (1 - cos(x))/2)

Simplifying the right-hand side, we get:

cos²(x/2) = (2cos(x) + 1 - cos(x))/2

cos²(x/2) = (cos(x) + 1)/2

Therefore, we have proved the half-angle formula for cosine:

cos²(x/2) = (1 + cos(x))/2.