Initial program 10.5
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
Initial simplification10.5
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\]
- Using strategy
rm Applied add-sqr-sqrt10.6
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\color{blue}{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}}\right)\]
Applied *-un-lft-identity10.6
\[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{1 \cdot \left(1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}}{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right)\]
Applied times-frac10.5
\[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}} \cdot \frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}}\right)\]
Applied sqrt-prod10.6
\[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}} \cdot \sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right)}\]
Final simplification10.6
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\sqrt{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2}}} \cdot \sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\sqrt{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2}}}\right)\]
