$\begin{align}  & \frac{1}{{{a}^{2}}}-\frac{1}{{{b}^{2}}}=\left( \frac{1}{a}-\frac{1}{b} \right)\left( \frac{1}{a}+\frac{1}{b} \right) \\ & \frac{1}{{{(1-\sqrt{3})}^{2}}}-\frac{1}{{{(1+\sqrt{3})}^{2}}}=\left( \frac{1}{(1-\sqrt{3})}-\frac{1}{(1+\sqrt{3})} \right)\left( \frac{1}{(1-\sqrt{3})}+\frac{1}{(1+\sqrt{3})} \right) \\ & \frac{1}{{{(1-\sqrt{3})}^{2}}}-\frac{1}{{{(1+\sqrt{3})}^{2}}}=\left[ \frac{1+\sqrt{3}-(1-\sqrt{3})}{(1-\sqrt{3})(1+\sqrt{3})} \right]\left[ \frac{(1+\sqrt{3})+(1-\sqrt{3})}{(1-\sqrt{3})(1+\sqrt{3})} \right] \\ & \frac{1}{{{(1-\sqrt{3})}^{2}}}-\frac{1}{{{(1+\sqrt{3})}^{2}}}=\left[ \frac{2\sqrt{3}}{1-3} \right]\left[ \frac{2}{1-3} \right]=\frac{4\sqrt{3}}{4}=\sqrt{3} \\\end{align}$