Initial program 32.4
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Initial simplification24.9
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
- Using strategy
rm Applied *-un-lft-identity24.9
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{1 \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}}\]
Applied times-frac18.4
\[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{1 \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]
Applied times-frac16.8
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{1} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 2}}\]
Simplified16.8
\[\leadsto \color{blue}{\frac{\frac{2}{t}}{\sin k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
- Using strategy
rm Applied add-sqr-sqrt16.9
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]
Applied *-un-lft-identity16.9
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}\]
Applied times-frac14.3
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}\]
Applied times-frac13.1
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{t}}{1}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}\right)}\]
Applied associate-*r*11.7
\[\leadsto \color{blue}{\left(\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]
- Using strategy
rm Applied add-cube-cbrt11.7
\[\leadsto \left(\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\color{blue}{\left(\sqrt[3]{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt[3]{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}\right) \cdot \sqrt[3]{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}}\]
Applied associate-/r*11.7
\[\leadsto \left(\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}\right) \cdot \color{blue}{\frac{\frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt[3]{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt[3]{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}}{\sqrt[3]{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}}\]
Final simplification11.7
\[\leadsto \left(\frac{\frac{\ell}{t}}{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \frac{\frac{2}{t}}{\sin k}\right) \cdot \frac{\frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt[3]{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \sqrt[3]{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}}}{\sqrt[3]{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}}\]
