Answer:
Using the half-angle formula for cosine, we can express cos(θ/2) in terms of the values of cos θ and sin θ:
cos(θ/2) = ±√[(1 + cos θ)/2]
The sign of the square root depends on the quadrant in which θ/2 lies.
Since we are given cos^2 θ/2, we need to eliminate the square root in the expression for cos(θ/2). To do this, we can use the fact that:
sin^2 θ/2 = 1 - cos^2 θ/2
Substituting the expression for cos^2 θ/2 from the previous identity, we get:
sin^2 θ/2 = 1 - cos^2 θ/2 = 1 - cos^2 (θ/2)
Solving for cos(θ/2) in terms of sin(θ/2), we get:
cos(θ/2) = ±√[1 - sin^2 θ/2]
Substituting the expression for sin^2 θ/2 from above, we get:
cos(θ/2) = ±√[cos^2 θ/2]
The sign of the square root depends on the quadrant in which θ/2 lies. Therefore, the completed identity for cos^2 θ/2 is:
cos(θ/2) = ±cos θ/2
Explanation: