Answer
$$\pm\sqrt{\frac{1+\cos18x}{2}}=\cos9x$$
Work Step by Step
$$\pm\sqrt{\frac{1+\cos18x}{2}}$$
Here we notice at the start of the expression, there is the sign $\pm$, which means we do not have to decide whether to take positive or negative square root.
From the half-angle identity for cosines:
$$\pm\sqrt{\frac{1+\cos A}{2}}=\cos\frac{A}{2}$$
We can apply the identity to the given expression with $A=18x$.
$$\pm\sqrt{\frac{1+\cos18x}{2}}=\cos\frac{18x}{2}$$
$$\pm\sqrt{\frac{1+\cos18x}{2}}=\cos9x$$
