Answer
$\displaystyle \cos\frac{x}{8}$
Work Step by Step
Comparing with the Half-Angle Identity,
$\displaystyle \cos\frac{A}{2}=\pm\sqrt{\frac{1+\cos A}{2}}$
if we replace $A$ with $\displaystyle \frac{x}{4}$ in the identity,
the RHS equals the given expression.
So, the LHS =$ \displaystyle \cos\frac{\frac{x}{4}}{2}=\cos\frac{x}{8}$