Initial program 9.8
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
Simplified9.8
\[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right)}\]
Taylor expanded around 0 21.0
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\color{blue}{\left(\frac{{t}^{2}}{{\ell}^{2}}\right)} \cdot 2 + 1)_*}}\right)\]
Simplified9.8
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot 2 + 1)_*}}\right)\]
- Using strategy
rm Applied *-un-lft-identity9.8
\[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{1 \cdot 1} - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right)\]
Applied difference-of-squares9.9
\[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\left(1 + \frac{Om}{Omc}\right) \cdot \left(1 - \frac{Om}{Omc}\right)}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right)\]
Applied associate-/l*9.9
\[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1 + \frac{Om}{Omc}}{\frac{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}{1 - \frac{Om}{Omc}}}}}\right)\]
- Using strategy
rm Applied add-cube-cbrt10.0
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1 + \frac{Om}{Omc}}{\frac{(\color{blue}{\left(\left(\sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}} \cdot \sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}}\right) \cdot \sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}}\right)} \cdot 2 + 1)_*}{1 - \frac{Om}{Omc}}}}\right)\]
Final simplification10.0
\[\leadsto \sin^{-1} \left(\sqrt{\frac{\frac{Om}{Omc} + 1}{\frac{(\left(\sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}} \cdot \left(\sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}} \cdot \sqrt[3]{\frac{t}{\ell} \cdot \frac{t}{\ell}}\right)\right) \cdot 2 + 1)_*}{1 - \frac{Om}{Omc}}}}\right)\]
