Initial program 47.0
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
Initial simplification30.7
\[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
- Using strategy
rm Applied times-frac29.8
\[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
Applied add-cube-cbrt29.9
\[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}}}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
Applied times-frac29.6
\[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
Applied times-frac18.0
\[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}{\frac{k}{t}}}\]
Simplified10.8
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \color{blue}{\left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)}\]
- Using strategy
rm Applied *-un-lft-identity10.8
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\color{blue}{1 \cdot \frac{k}{t}}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied div-inv10.9
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\color{blue}{\sin k \cdot \frac{1}{\frac{\ell}{t}}}}}{1 \cdot \frac{k}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied times-frac10.9
\[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{1}{\frac{\ell}{t}}}}}{1 \cdot \frac{k}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied times-frac10.9
\[\leadsto \color{blue}{\left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}}{1} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{1}{\frac{\ell}{t}}}}{\frac{k}{t}}\right)} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Simplified10.9
\[\leadsto \left(\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{1}{\frac{\ell}{t}}}}{\frac{k}{t}}\right) \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Simplified7.1
\[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k} \cdot \color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}}}\right) \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
- Using strategy
rm Applied div-inv7.1
\[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}}\right) \cdot \left(\left(\frac{1}{k} \cdot \color{blue}{\left(\ell \cdot \frac{1}{t}\right)}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied associate-*r*2.3
\[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}}\right) \cdot \left(\color{blue}{\left(\left(\frac{1}{k} \cdot \ell\right) \cdot \frac{1}{t}\right)} \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
- Using strategy
rm Applied tan-quot2.3
\[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k} \cdot \frac{\sqrt[3]{\frac{2}{\color{blue}{\frac{\sin k}{\cos k}}}}}{\frac{k}{\ell}}\right) \cdot \left(\left(\left(\frac{1}{k} \cdot \ell\right) \cdot \frac{1}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied associate-/r/2.3
\[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k} \cdot \frac{\sqrt[3]{\color{blue}{\frac{2}{\sin k} \cdot \cos k}}}{\frac{k}{\ell}}\right) \cdot \left(\left(\left(\frac{1}{k} \cdot \ell\right) \cdot \frac{1}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied cbrt-prod2.3
\[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k} \cdot \frac{\color{blue}{\sqrt[3]{\frac{2}{\sin k}} \cdot \sqrt[3]{\cos k}}}{\frac{k}{\ell}}\right) \cdot \left(\left(\left(\frac{1}{k} \cdot \ell\right) \cdot \frac{1}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Final simplification2.3
\[\leadsto \left(\left(\frac{1}{t} \cdot \left(\ell \cdot \frac{1}{k}\right)\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \left(\frac{\sqrt[3]{\frac{2}{\sin k}} \cdot \sqrt[3]{\cos k}}{\frac{k}{\ell}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}\right)\]
