Initial program 32.2
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Initial simplification24.9
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
- Using strategy
rm Applied *-un-lft-identity24.9
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{1 \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Applied times-frac18.9
\[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{1 \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
Applied times-frac17.2
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{1} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Simplified17.2
\[\leadsto \color{blue}{\frac{\frac{2}{t}}{\sin k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
- Using strategy
rm Applied add-sqr-sqrt17.3
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\]
Applied *-un-lft-identity17.3
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Applied times-frac14.5
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Applied times-frac13.2
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{t}}{1}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\right)}\]
Applied associate-*r*11.8
\[\leadsto \color{blue}{\left(\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\]
- Using strategy
rm Applied add-cube-cbrt11.9
\[\leadsto \left(\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{\color{blue}{\left(\sqrt[3]{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}} \cdot \sqrt[3]{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\right) \cdot \sqrt[3]{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Applied div-inv11.9
\[\leadsto \left(\frac{\frac{2}{t}}{\sin k} \cdot \frac{\color{blue}{\frac{\ell}{t} \cdot \frac{1}{1}}}{\left(\sqrt[3]{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}} \cdot \sqrt[3]{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\right) \cdot \sqrt[3]{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Applied times-frac11.9
\[\leadsto \left(\frac{\frac{2}{t}}{\sin k} \cdot \color{blue}{\left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}} \cdot \sqrt[3]{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}} \cdot \frac{\frac{1}{1}}{\sqrt[3]{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right)}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Simplified11.9
\[\leadsto \left(\frac{\frac{2}{t}}{\sin k} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}} \cdot \sqrt[3]{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}} \cdot \color{blue}{\frac{1}{\sqrt[3]{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}}\right)\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Final simplification11.9
\[\leadsto \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}} \cdot \left(\left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}} \cdot \sqrt[3]{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}} \cdot \frac{1}{\sqrt[3]{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right) \cdot \frac{\frac{2}{t}}{\sin k}\right)\]
