Initial program 10.3
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
Initial simplification10.3
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\]
- Using strategy
rm Applied add-cube-cbrt10.4
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}} \cdot \sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}}\right) \cdot \sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}}\right)}}}\right)\]
- Using strategy
rm Applied add-sqr-sqrt10.4
\[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \left(\left(\sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}} \cdot \sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}}\right) \cdot \sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}}\right)}} \cdot \sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \left(\left(\sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}} \cdot \sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}}\right) \cdot \sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}}\right)}}}}\right)\]
Applied rem-sqrt-square10.4
\[\leadsto \sin^{-1} \color{blue}{\left(\left|\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \left(\left(\sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}} \cdot \sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}}\right) \cdot \sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}}\right)}}\right|\right)}\]
Simplified10.3
\[\leadsto \sin^{-1} \left(\left|\color{blue}{\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1}}}\right|\right)\]
Final simplification10.3
\[\leadsto \sin^{-1} \left(\left|\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2}}\right|\right)\]
