Initial program 10.4
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
Initial simplification10.4
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right)\]
- Using strategy
rm Applied expm1-log1p-u10.4
\[\leadsto \color{blue}{(e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right))} - 1)^*}\]
- Using strategy
rm Applied add-sqr-sqrt10.4
\[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\color{blue}{\sqrt{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*} \cdot \sqrt{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}}}\right))} - 1)^*\]
Applied associate-/r*10.4
\[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\sqrt{\color{blue}{\frac{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\sqrt{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}}{\sqrt{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}}}\right))} - 1)^*\]
- Using strategy
rm Applied sqrt-div10.5
\[\leadsto (e^{\log_* (1 + \sin^{-1} \color{blue}{\left(\frac{\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\sqrt{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}}}{\sqrt{\sqrt{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}}\right)})} - 1)^*\]
Final simplification10.5
\[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\frac{\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\sqrt{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}}}{\sqrt{\sqrt{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}}\right))} - 1)^*\]
