Find the Exact Value by Using a Half-Angle Identity $\sin 22.5 ^ { \circ }$

Find the exact value by using a half-angle identity.
- $\sin 22.5 ^ { \circ }$

A) $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 2 } }$
B) $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 2 } }$
C) $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 2 } }$
D) $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 2 } }$