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 clear-num10.8
\[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{\frac{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}}\right)\]
- Using strategy
rm Applied *-un-lft-identity10.8
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\frac{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}{\color{blue}{1 \cdot \left(1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}}}}\right)\]
Applied add-sqr-sqrt10.8
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\frac{\color{blue}{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}{1 \cdot \left(1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}}}\right)\]
Applied times-frac10.8
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\frac{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}{1} \cdot \frac{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}}\right)\]
Applied associate-/r*10.8
\[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\frac{1}{\frac{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}{1}}}{\frac{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}}\right)\]
- Using strategy
rm Applied flip3--10.8
\[\leadsto \sin^{-1} \left(\sqrt{\frac{\frac{1}{\frac{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}{1}}}{\frac{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}{\color{blue}{\frac{{1}^{3} - {\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}^{3}}{1 \cdot 1 + \left(\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) + 1 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)}}}}}\right)\]
Applied associate-/r/10.8
\[\leadsto \sin^{-1} \left(\sqrt{\frac{\frac{1}{\frac{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}{1}}}{\color{blue}{\frac{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}{{1}^{3} - {\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) + 1 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\right)}}}\right)\]
Applied associate-/r/10.8
\[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\frac{1}{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}} \cdot 1}}{\frac{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}{{1}^{3} - {\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) + 1 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\right)}}\right)\]
Applied times-frac10.8
\[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\frac{1}{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}{\frac{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}{{1}^{3} - {\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}^{3}}} \cdot \frac{1}{1 \cdot 1 + \left(\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) + 1 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)}}}\right)\]
Applied sqrt-prod10.8
\[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{\frac{1}{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}{\frac{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}{{1}^{3} - {\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}^{3}}}} \cdot \sqrt{\frac{1}{1 \cdot 1 + \left(\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) + 1 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)}}\right)}\]
Simplified10.8
\[\leadsto \sin^{-1} \left(\color{blue}{\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{5} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}} \cdot \sqrt{\frac{1}{1 \cdot 1 + \left(\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) + 1 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)}}\right)\]
Final simplification10.8
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{1 + \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc} + \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)}} \cdot \sqrt{\frac{1 - \frac{Om}{Omc} \cdot {\left(\frac{Om}{Omc}\right)}^{5}}{\frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + 1}}\right)\]
