Rationalize the denominator: \(\frac{1+2sinA}{\sqrt{3}-2cosA} \cdot \frac{\sqrt{3}+2cosA}{\sqrt{3}+2cosA}=\frac{\sqrt{3}+2cosA+2sinA\sqrt{3}+4sinAcosA}{{\sqrt{3}}^{2}-{(2cosA)}^{2}}\).
\[{(\frac{\sqrt{3}+2cosA}{1-2sinA})}^{-3}+{(\frac{\sqrt{3}+2cosA+2sinA\sqrt{3}+4sinAcosA}{{\sqrt{3}}^{2}-{(2cosA)}^{2}})}^{-3}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[{(\frac{\sqrt{3}+2cosA}{1-2sinA})}^{-3}+{(\frac{\sqrt{3}+2cosA+2sinA\sqrt{3}+4sinAcosA}{3-{(2cosA)}^{2}})}^{-3}\]
Convert to common denominators.
\[{(\frac{\sqrt{3}+2cosA}{1-2sinA})}^{-3}+{(\frac{\sqrt{3}+2cosA+2sinA\sqrt{3}+4sinAcosA}{3-4{cosA}^{2}})}^{-3}\]
Use Negative Power Rule: \({x}^{-a}=\frac{1}{{x}^{a}}\).
\[\frac{1}{{(\frac{\sqrt{3}+2cosA}{1-2sinA})}^{3}}+{(\frac{\sqrt{3}+2cosA+2sinA\sqrt{3}+4sinAcosA}{3-4{cosA}^{2}})}^{-3}\]
Use Division Distributive Property: \({(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}\).
\[\frac{1}{\frac{{(\sqrt{3}+2cosA)}^{3}}{{(1-2sinA)}^{3}}}+{(\frac{\sqrt{3}+2cosA+2sinA\sqrt{3}+4sinAcosA}{3-4{cosA}^{2}})}^{-3}\]
Use Negative Power Rule: \({x}^{-a}=\frac{1}{{x}^{a}}\).
\[\frac{1}{\frac{{(\sqrt{3}+2cosA)}^{3}}{{(1-2sinA)}^{3}}}+\frac{1}{{(\frac{\sqrt{3}+2cosA+2sinA\sqrt{3}+4sinAcosA}{3-4{cosA}^{2}})}^{3}}\]
Use Division Distributive Property: \({(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}\).
\[\frac{1}{\frac{{(\sqrt{3}+2cosA)}^{3}}{{(1-2sinA)}^{3}}}+\frac{1}{\frac{{(\sqrt{3}+2cosA+2sinA\sqrt{3}+4sinAcosA)}^{3}}{{(3-4{cosA}^{2})}^{3}}}\]
Invert and multiply.
\[\frac{{(1-2sinA)}^{3}}{{(\sqrt{3}+2cosA)}^{3}}+\frac{1}{\frac{{(\sqrt{3}+2cosA+2sinA\sqrt{3}+4sinAcosA)}^{3}}{{(3-4{cosA}^{2})}^{3}}}\]
Invert and multiply.
\[\frac{{(1-2sinA)}^{3}}{{(\sqrt{3}+2cosA)}^{3}}+\frac{{(3-4{cosA}^{2})}^{3}}{{(\sqrt{3}+2cosA+2sinA\sqrt{3}+4sinAcosA)}^{3}}\]
Decimal Form: 5.388603
(1-2*sinA)^3/(sqrt(3)+2*cosA)^3+(3-4*cosA^2)^3/(sqrt(3)+2*cosA+2*sinA*sqrt(3)+4*sinA*cosA)^3
