Chapter 7 - Trigonometric Identities and Equations - 7.4 Double-Angle and Half-Angle Identities - 7.4 Exercises - Page 694: 77

$\displaystyle \cos\frac{x}{8}$

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}$