Simplify $\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}-\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$.

Let's find a common denominator to subtract the two fractions:
$$\text{Expression}=\frac{(\sqrt{5}-\sqrt{3}{)}^2-(\sqrt{5}+\sqrt{3}{)}^2}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})}$$
Step 1: Simplify the denominatorUsing the difference of squares identity $(a+b)(a-b)=a^2-b^2$:
$$\text{Denominator}=(\sqrt{5}{)}^2-(\sqrt{3}{)}^2=5-3=2$$
Step 2: Simplify the numeratorUsing the algebraic identity $(x-y{)}^2-(x+y{)}^2=-4xy$:
$$\text{Numerator}=-4(\sqrt{5})(\sqrt{3})=-4\sqrt{15}$$
Step 3: Combine numerator and denominator
$$\text{Expression}=\frac{-4\sqrt{15}}{2}=-2\sqrt{15}$$