Answer
$$\cos\frac{x}{2}=\frac{\sqrt{10}}{4}$$
Work Step by Step
$$\cos x=\frac{1}{4}\hspace{1.5cm}0\lt x\lt\frac{\pi}{2}\hspace{1.5cm}\cos\frac{x}{2}=?$$
Apply the half-angle identity for cosine
$$\cos\frac{x}{2}=\pm\sqrt{\frac{1+\cos x}{2}}$$
As $0\lt x\lt\frac{\pi}{2}$, we deduce that $0\lt\frac{x}{2}\lt\frac{\pi}{4}$.
That means the angle $\frac{x}{2}$ lies in quadrant I, where cosines are positive. That means $\cos\frac{x}{2}\gt0$, so we need to choose the positive square root.
$$\cos\frac{x}{2}=\sqrt{\frac{1+\cos x}{2}}$$
$$\cos\frac{x}{2}=\sqrt{\frac{1+\frac{1}{4}}{2}}$$
$$\cos\frac{x}{2}=\sqrt{\frac{\frac{5}{4}}{2}}$$
$$\cos\frac{x}{2}=\sqrt{\frac{5}{8}}$$
$$\cos\frac{x}{2}=\frac{\sqrt5}{2\sqrt2}$$
$$\cos\frac{x}{2}=\frac{\sqrt{5}\times\sqrt2}{2\times2}$$
$$\cos\frac{x}{2}=\frac{\sqrt{10}}{4}$$
