Answer:
cos^2 θ/2 = (cos θ + 1)/2.
Explanation:
We can use the identity cos 2θ = 2cos^2 θ - 1 to derive an identity for cos^2 θ/2:
cos 2(θ/2) = cos θ
Replacing θ/2 with x:
cos 2x = cos 2(x/2)
Using the double angle formula for cosine:
cos 2x = 2cos^2 x - 1
Substituting θ/2 for x:
cos θ = 2cos^2 (θ/2) - 1
Rearranging this formula, we can solve for cos^2 θ/2:
2cos^2 (θ/2) = cos θ + 1
cos^2 (θ/2) = (cos θ + 1)/2
Therefore, cos^2 θ/2 = (cos θ + 1)/2.