$\begin{align}
& \frac{1}{\sqrt{49+20\sqrt{6}}}-\frac{1}{\sqrt{49-20\sqrt{6}}}+\frac{1}{\sqrt{7-4\sqrt{3}}} \\
& =-4\sqrt{6}+\frac{1}{2-\sqrt{3}} \\
& =-4\sqrt{6}+\frac{1.\left( 2+\sqrt{3} \right)}{4-3}=-4\sqrt{6}+2+\sqrt{3}. \\
\end{align}$

$\frac{1}{\sqrt{49+20\sqrt{6}}}-\frac{1}{\sqrt{49-20\sqrt{6}}}+\frac{1}{\sqrt{7-4\sqrt{3}}}$
$=\frac{1}{\sqrt{25+2.5.2\sqrt{6}+24}}-\frac{1}{\sqrt{25-2.5.2\sqrt{6}+24}}+\frac{1}{\sqrt{4-2.2.\sqrt{3}+3}}$
$=\frac{1}{\sqrt{{{\left( 5+2\sqrt{6} \right)}^{2}}}}-\frac{1}{\sqrt{{{\left( 5-2\sqrt{6} \right)}^{2}}}}+\frac{1}{\sqrt{{{\left( 2-\sqrt{3} \right)}^{2}}}}$
$=\frac{1}{5+2\sqrt{6}}-\frac{1}{5-2\sqrt{6}}+\frac{1}{2-\sqrt{3}}$
$=\frac{5-2\sqrt{6}-\left( 5+2\sqrt{6} \right)}{\left( 5+2\sqrt{6} \right)\left( 5-2\sqrt{6} \right)}+\frac{1}{2-\sqrt{3}}$
$=-4\sqrt{6}+\frac{1}{2-\sqrt{3}}$