Initial program 47.2
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
Initial simplification30.2
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 0}\]
- Using strategy
rm Applied add-sqr-sqrt30.2
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}}\]
Applied times-frac30.1
\[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]
Applied times-frac27.2
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}}\]
Simplified27.2
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]
Simplified13.0
\[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right)}\]
- Using strategy
rm Applied add-cube-cbrt13.3
\[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{\left(\sqrt[3]{\left|\frac{k}{t}\right|} \cdot \sqrt[3]{\left|\frac{k}{t}\right|}\right) \cdot \sqrt[3]{\left|\frac{k}{t}\right|}}} \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right)\]
Applied *-un-lft-identity13.3
\[\leadsto \frac{\frac{\frac{2}{t}}{\color{blue}{1 \cdot \sin k}}}{\left(\sqrt[3]{\left|\frac{k}{t}\right|} \cdot \sqrt[3]{\left|\frac{k}{t}\right|}\right) \cdot \sqrt[3]{\left|\frac{k}{t}\right|}} \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right)\]
Applied add-cube-cbrt13.3
\[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}\right) \cdot \sqrt[3]{\frac{2}{t}}}}{1 \cdot \sin k}}{\left(\sqrt[3]{\left|\frac{k}{t}\right|} \cdot \sqrt[3]{\left|\frac{k}{t}\right|}\right) \cdot \sqrt[3]{\left|\frac{k}{t}\right|}} \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right)\]
Applied times-frac13.3
\[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sin k}}}{\left(\sqrt[3]{\left|\frac{k}{t}\right|} \cdot \sqrt[3]{\left|\frac{k}{t}\right|}\right) \cdot \sqrt[3]{\left|\frac{k}{t}\right|}} \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right)\]
Applied times-frac12.6
\[\leadsto \color{blue}{\left(\frac{\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}}{\sqrt[3]{\left|\frac{k}{t}\right|} \cdot \sqrt[3]{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sin k}}{\sqrt[3]{\left|\frac{k}{t}\right|}}\right)} \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right)\]
Simplified12.6
\[\leadsto \left(\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\left|\frac{k}{t}\right|}} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\left|\frac{k}{t}\right|}}\right)} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sin k}}{\sqrt[3]{\left|\frac{k}{t}\right|}}\right) \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right)\]
Final simplification12.6
\[\leadsto \left(\left(\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\left|\frac{k}{t}\right|}} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\left|\frac{k}{t}\right|}}\right) \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sin k}}{\sqrt[3]{\left|\frac{k}{t}\right|}}\right) \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right)\]
