$latex \mathbf{2\sqrt{2}}$
$latex \because \sqrt{5+2\sqrt{6}} = \sqrt{3+2+2\times \sqrt{3}\times \sqrt{2}}$
$latex = \sqrt{(\sqrt{3}+\sqrt{2})^{2}}$
$latex = \sqrt{3}+\sqrt{2}$
$latex \therefore \frac{1}{\sqrt{5+2\sqrt{6}}} = \sqrt{3}-\sqrt{2}&s=1$
$latex \therefore \sqrt{5+2\sqrt{6}}-\frac{1}{\sqrt{5+2\sqrt{6}}} = \sqrt{3}+\sqrt{2}-\sqrt{3}+\sqrt{2}= 2\sqrt{2}$
Hence option [C] is correct answer.