Using the double-angle identities, show how to derive the half-angle identity for \( \cos \left(\frac{x}{2}\right) \).

To derive the half-angle identity for \( \cos \left(\frac{x}{2}\right) \), we can start with the double-angle identity for cosine, which states: \[ \cos(2\theta) = 2\cos^2(\theta) - 1 \] If we set \( \theta = \frac{x}{2} \), then \( 2\theta = x \). Replacing \( \theta \) in the double-angle identity gives us: \[ \cos(x) = 2\cos^2\left(\frac{x}{2}\right) - 1 \] Next, we can rearrange this equation to solve for \( \cos^2\left(\frac{x}{2}\right) \): \[ \cos^2\left(\frac{x}{2}\right) = \frac{1 + \cos(x)}{2} \] To find the half-angle identity, we take the square root, giving us: \[ \cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos(x)}{2}} \] The sign depends on the quadrant in which \( \frac{x}{2} \) lies, but this result is our half-angle identity for cosine.