${\left(\frac{\sqrt{3}+\mathrm{cosA}}{1-2\mathrm{sinA}}\right)}^{-3}+{\left(\frac{1+\mathrm{sinA}}{\sqrt{3}-2\mathrm{cosA}}\right)}^{-3}=$

$\begin{array}{rcl} & & {\left(\frac{\sqrt{3}+2\cos \mathrm{A}}{1-2 \mathrm{sinA}}\right)}^{-3}+{\left(\frac{1+2\sin \mathrm{A}}{\sqrt{3}-2 \cos \mathrm{A}}\right)}^{-3}\\ & =& {\left(\frac{1-2 \mathrm{sinA}}{\sqrt{3}+2\mathrm{cosA}}\right)}^3+{\left(\frac{\sqrt{3}-2\mathrm{cosA}}{1+2\mathrm{sinA}}\right)}^3\\ & =& \frac{{\left(1-4{\sin }^2\mathrm{A}\right)}^3+{\left(3-4{\cos }^2\mathrm{A}\right)}^3}{{\left(\sqrt{3}+2\mathrm{cosA}\right)}^3{\left(1+2\mathrm{sinA}\right)}^3}\\ & =& \frac{{\left(1-4{\sin }^2\mathrm{A}\right)}^3-{\left(1-4{\sin }^2\mathrm{A}\right)}^3}{{\left(\sqrt{3}+2\mathrm{cosA}\right)}^3{\left(1+2\mathrm{sinA}\right)}^3}\\ & =& 0\end{array}$