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 simplification31.2
\[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
- Using strategy
rm Applied times-frac30.1
\[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
Applied add-cube-cbrt30.2
\[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}}}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
Applied times-frac29.8
\[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
Applied times-frac18.3
\[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}{\frac{k}{t}}}\]
Simplified11.0
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \color{blue}{\left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)}\]
- Using strategy
rm Applied div-inv11.0
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\color{blue}{k \cdot \frac{1}{t}}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied *-un-lft-identity11.0
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\color{blue}{1 \cdot \frac{\ell}{t}}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied *-un-lft-identity11.0
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\color{blue}{1 \cdot \sin k}}{1 \cdot \frac{\ell}{t}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied times-frac11.0
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\color{blue}{\frac{1}{1} \cdot \frac{\sin k}{\frac{\ell}{t}}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied times-frac11.0
\[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied times-frac7.2
\[\leadsto \color{blue}{\left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{1}{1}}}{k} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{1}{t}}\right)} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Simplified7.2
\[\leadsto \left(\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{k}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{1}{t}}\right) \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Simplified7.2
\[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{k} \cdot \color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\ell}}}\right) \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
- Using strategy
rm Applied associate-*r/2.4
\[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{k} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\ell}}\right) \cdot \left(\color{blue}{\frac{\frac{1}{k} \cdot \ell}{t}} \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied associate-*l/2.0
\[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{k} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\ell}}\right) \cdot \color{blue}{\frac{\left(\frac{1}{k} \cdot \ell\right) \cdot \sqrt[3]{\frac{2}{\tan k}}}{t}}\]
Simplified2.0
\[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{k} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\ell}}\right) \cdot \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}}}}{t}\]
- Using strategy
rm Applied tan-quot1.9
\[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{k} \cdot \frac{\sqrt[3]{\frac{2}{\color{blue}{\frac{\sin k}{\cos k}}}}}{\frac{\sin k}{\ell}}\right) \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}}}{t}\]
Applied associate-/r/1.9
\[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{k} \cdot \frac{\sqrt[3]{\color{blue}{\frac{2}{\sin k} \cdot \cos k}}}{\frac{\sin k}{\ell}}\right) \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}}}{t}\]
Applied cbrt-prod1.9
\[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{k} \cdot \frac{\color{blue}{\sqrt[3]{\frac{2}{\sin k}} \cdot \sqrt[3]{\cos k}}}{\frac{\sin k}{\ell}}\right) \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}}}{t}\]
Final simplification1.9
\[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\sin k}} \cdot \sqrt[3]{\cos k}}{\frac{\sin k}{\ell}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{k}\right) \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}}}{t}\]
