Explanation:
The correct answer is C. \(\sqrt{2} - 1\).

Explanation:

Using a half-angle identity for tangent:

\[
\tan \left( \frac{\theta}{2} \right) = \frac{1 - \cos \theta}{\sin \theta}
\]

For \(\theta = 45^\circ\), we have:

\[
\tan \left( 22.5^\circ \right) = \sqrt{2} - 1
\]

Thus, the surd form of \(\tan 22 \frac{1}{2}^\circ\) is \(\sqrt{2} - 1\). Therefore, option C is correct.