Answer
$$\pm\sqrt{\frac{1+\cos\frac{x}{4}}{2}}=\cos\frac{x}{8}$$
Work Step by Step
$$\pm\sqrt{\frac{1+\cos\frac{x}{4}}{2}}$$
From the half-angle identity for cosine:
$$\pm\sqrt{\frac{1+\cos\theta}{2}}=\cos\frac{\theta}{2}$$
We can apply the identity to the given expression with $\theta=\frac{x}{4}$.
$$\pm\sqrt{\frac{1+\cos\frac{x}{4}}{2}}=\cos\frac{\frac{x}{4}}{2}$$
$$\pm\sqrt{\frac{1+\cos\frac{x}{4}}{2}}=\cos\frac{x}{8}$$
