Why must x be acute for the identity \sqrt{\frac{1-\sin x}{1+\sin

The important point is that “identity” is true only when ${\displaystyle -{90}^{\circ }<x<{90}^{\circ }}$.
In another word, it is not true when the terminal arm falls in QII and QIII.
Note that the angle lies in ${\displaystyle -{90}^{\circ }<x<0^{\circ }}$ is also an acute angle, because the minus sign only refers to moving the terminal arm in the clockwise direction. The resultant angle is still acute.
Added: To move the terminal arm through ${\displaystyle {330}^{\circ }}$ in the anticlockwise direction is the same as moving it through ${\displaystyle {30}^{\circ }}$ in the clockwise direction.