Answer
$-\frac{\sqrt {2-\sqrt 2}}{2}$
Work Step by Step
With $22.5^\circ=\frac{45^\circ}{2}$, use the half-angle identity, we have $sin(-22.5^\circ)=-sin(22.5^\circ)=-\sqrt {\frac{1-cos(45^\circ)}{2}}=-\sqrt {\frac{1-\frac{\sqrt 2}{2}}{2}}=-\frac{\sqrt {2-\sqrt 2}}{2}$
