Initial program 10.5
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
- Using strategy
rm Applied sqrt-div10.6
\[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)}\]
- Using strategy
rm Applied add-sqr-sqrt10.6
\[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\color{blue}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right)\]
Applied rem-sqrt-square10.6
\[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\left|\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right|}}\right)\]
Simplified1.2
\[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\left|\color{blue}{\sqrt{1^2 + \left(\sqrt{2} \cdot \frac{t}{\ell}\right)^2}^*}\right|}\right)\]
- Using strategy
rm Applied div-inv1.2
\[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\left|\sqrt{1^2 + \left(\sqrt{2} \cdot \color{blue}{\left(t \cdot \frac{1}{\ell}\right)}\right)^2}^*\right|}\right)\]
Applied associate-*r*1.2
\[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\left|\sqrt{1^2 + \color{blue}{\left(\left(\sqrt{2} \cdot t\right) \cdot \frac{1}{\ell}\right)}^2}^*\right|}\right)\]
Final simplification1.2
\[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\left|\sqrt{1^2 + \left(\frac{1}{\ell} \cdot \left(\sqrt{2} \cdot t\right)\right)^2}^*\right|}\right)\]
