Initial program 10.2
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
Initial simplification10.2
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\]
- Using strategy
rm Applied flip--10.2
\[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\frac{1 \cdot 1 - \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}{1 + \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\]
Applied associate-/l/10.2
\[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1 \cdot 1 - \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}{\left(1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(1 + \frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}}}\right)\]
Simplified10.2
\[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{1 - {\left(\frac{Om}{Omc}\right)}^{4}}}{\left(1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(1 + \frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}}\right)\]
Final simplification10.2
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{4}}{\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc} + 1\right)}}\right)\]
