We are given the expression $\csc 10^{\circ} - \sqrt{3} \sec 10^{\circ}$.

We can rewrite this expression in terms of sine and cosine:

$\frac{1}{\sin 10^{\circ}} - \frac{\sqrt{3}}{\cos 10^{\circ}}$

Now, we find a common denominator:

$\frac{\cos 10^{\circ} - \sqrt{3} \sin 10^{\circ}}{\sin 10^{\circ} \cos 10^{\circ}}$

Multiply and divide by 2:

$\frac{2(\frac{1}{2} \cos 10^{\circ} - \frac{\sqrt{3}}{2} \sin 10^{\circ})}{\sin 10^{\circ} \cos 10^{\circ}}$

Recognize that $\cos 60^{\circ} = \frac{1}{2}$ and $\sin 60^{\circ} = \frac{\sqrt{3}}{2}$:

$\frac{2(\cos 60^{\circ} \cos 10^{\circ} - \sin 60^{\circ} \sin 10^{\circ})}{\sin 10^{\circ} \cos 10^{\circ}}$

Use the cosine addition formula $\cos(A+B) = \cos A \cos B - \sin A \sin B$:

$\frac{2 \cos (60^{\circ} + 10^{\circ})}{\sin 10^{\circ} \cos 10^{\circ}} = \frac{2 \cos 70^{\circ}}{\sin 10^{\circ} \cos 10^{\circ}}$

Since $\cos 70^{\circ} = \sin 20^{\circ}$, we have:

$\frac{2 \sin 20^{\circ}}{\sin 10^{\circ} \cos 10^{\circ}}$

Multiply and divide by 2 again:

$\frac{4 \sin 20^{\circ}}{2 \sin 10^{\circ} \cos 10^{\circ}}$

Use the sine double angle formula $2 \sin A \cos A = \sin 2A$:

$\frac{4 \sin 20^{\circ}}{\sin 20^{\circ}}$

Simplify:

$4$