Answer
The positive square root should be selected.
$$\cos58^\circ=\sqrt{\frac{1+\cos116^\circ}{2}}$$
Work Step by Step
$$\cos58^\circ=\pm\sqrt{\frac{1+\cos116^\circ}{2}}$$
Whether the positive or negative square root should be selected depends the sign of $\cos58^\circ$.
$58^\circ$ lies in quadrant I. In quadrant I, $\cos\theta\gt0$. Therefore, $\cos58^\circ\gt0$.
Thus, the positive square root should be selected. In other words,
$$\cos58^\circ=\sqrt{\frac{1+\cos116^\circ}{2}}$$