#### Answer

The negative square root should be selected.
$$\sin(-10^\circ)=-\sqrt{\frac{1-\cos(-20^\circ)}{2}}$$

#### Work Step by Step

$$\sin(-10^\circ)=\pm\sqrt{\frac{1-\cos(-20^\circ)}{2}}$$
Whether the positive or negative square root should be selected depends the sign of $\sin(-10^\circ)$.
The position of angle $-10^\circ$ collides with that of angle $350^\circ$. So we can consider $\sin(-10^\circ)=\sin350^\circ$
$350^\circ$ lies in quadrant IV. In quadrant IV, $\sin\theta\lt0$. Thus, $\sin350^\circ=\sin(-10^\circ)\lt0$.
Thus, the negative square root should be selected. In other words,
$$\sin(-10^\circ)=-\sqrt{\frac{1-\cos(-20^\circ)}{2}}$$