Answer
$\pm \sqrt{\dfrac{1-\cos{x}}{2}}$
Work Step by Step
Using the double angle formula:
$$\cos{2A} = 1 - 2 \sin^2{A}$$
$$2 \sin^2{A}= 1-\cos{2A}$$
$$\sin^2{A} = \dfrac{1-\cos{2A}}{2}$$
$$\therefore \sin{A} = \pm \sqrt{\dfrac{1-\cos{2A}}{2}}$$
Replacing $A$ with $\dfrac{x}{2}$:
$\sin{\dfrac{x}{2}} = \pm \sqrt{\dfrac{1-\cos{x}}{2}}$
