Initial program 10.7
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
Initial simplification10.7
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\]
- Using strategy
rm Applied flip--10.7
\[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\frac{1 \cdot 1 - \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}{1 + \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\]
Applied associate-/l/10.7
\[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1 \cdot 1 - \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}{\left(1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(1 + \frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}}}\right)\]
Simplified10.7
\[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{1 - {\left(\frac{Om}{Omc}\right)}^{4}}}{\left(1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(1 + \frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}}\right)\]
- Using strategy
rm Applied flip-+10.8
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{4}}{\left(1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \color{blue}{\frac{1 \cdot 1 - \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}}\right)\]
Applied associate-*r/10.8
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{4}}{\color{blue}{\frac{\left(1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(1 \cdot 1 - \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)}{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}}\right)\]
Applied associate-/r/10.7
\[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{4}}{\left(1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(1 \cdot 1 - \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)} \cdot \left(1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}}\right)\]
Applied sqrt-prod10.7
\[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{4}}{\left(1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(1 \cdot 1 - \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)}} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}\]
Simplified10.8
\[\leadsto \sin^{-1} \left(\color{blue}{\sqrt{\frac{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{4}}{1 - {\left(\frac{Om}{Omc}\right)}^{4}}}{\frac{\frac{2}{\frac{\ell}{t}}}{\frac{\ell}{t}} + 1}}} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\]
- Using strategy
rm Applied div-inv10.7
\[\leadsto \sin^{-1} \left(\sqrt{\frac{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{4}}{1 - {\left(\frac{Om}{Omc}\right)}^{4}}}{\frac{\frac{2}{\color{blue}{\ell \cdot \frac{1}{t}}}}{\frac{\ell}{t}} + 1}} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\]
Applied associate-/r*10.8
\[\leadsto \sin^{-1} \left(\sqrt{\frac{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{4}}{1 - {\left(\frac{Om}{Omc}\right)}^{4}}}{\frac{\color{blue}{\frac{\frac{2}{\ell}}{\frac{1}{t}}}}{\frac{\ell}{t}} + 1}} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\]
Final simplification10.8
\[\leadsto \sin^{-1} \left(\sqrt{\frac{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{4}}{1 - {\left(\frac{Om}{Omc}\right)}^{4}}}{\frac{\frac{\frac{2}{\ell}}{\frac{1}{t}}}{\frac{\ell}{t}} + 1}} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\]
