Initial program 47.3
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
Initial simplification30.3
\[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
- Using strategy
rm Applied times-frac29.3
\[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
Applied add-cube-cbrt29.4
\[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}}}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
Applied times-frac29.0
\[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
Applied times-frac18.2
\[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}{\frac{k}{t}}}\]
Simplified10.9
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \color{blue}{\left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)}\]
- Using strategy
rm Applied div-inv10.9
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\color{blue}{k \cdot \frac{1}{t}}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied *-un-lft-identity10.9
\[\leadsto \frac{\color{blue}{1 \cdot \frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied times-frac6.9
\[\leadsto \color{blue}{\left(\frac{1}{k} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{1}{t}}\right)} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Simplified6.8
\[\leadsto \left(\frac{1}{k} \cdot \color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\frac{\sin k}{\ell}}{\sqrt[3]{\frac{2}{\tan k}}}}}\right) \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
- Using strategy
rm Applied div-inv6.8
\[\leadsto \left(\frac{1}{k} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\frac{\sin k}{\ell}}{\sqrt[3]{\frac{2}{\tan k}}}}\right) \cdot \left(\left(\frac{1}{k} \cdot \color{blue}{\left(\ell \cdot \frac{1}{t}\right)}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied associate-*r*2.2
\[\leadsto \left(\frac{1}{k} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\frac{\sin k}{\ell}}{\sqrt[3]{\frac{2}{\tan k}}}}\right) \cdot \left(\color{blue}{\left(\left(\frac{1}{k} \cdot \ell\right) \cdot \frac{1}{t}\right)} \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Taylor expanded around -inf 32.5
\[\leadsto \left(\frac{1}{k} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\frac{\sin k}{\ell}}{\sqrt[3]{\frac{2}{\tan k}}}}\right) \cdot \left(\left(\left(\frac{1}{k} \cdot \ell\right) \cdot \frac{1}{t}\right) \cdot \color{blue}{{\left(2 \cdot \frac{\cos k}{\sin k}\right)}^{\frac{1}{3}}}\right)\]
Simplified2.2
\[\leadsto \left(\frac{1}{k} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\frac{\sin k}{\ell}}{\sqrt[3]{\frac{2}{\tan k}}}}\right) \cdot \left(\left(\left(\frac{1}{k} \cdot \ell\right) \cdot \frac{1}{t}\right) \cdot \color{blue}{\sqrt[3]{\frac{\cos k}{\sin k} \cdot 2}}\right)\]
Final simplification2.2
\[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\frac{\sin k}{\ell}}{\sqrt[3]{\frac{2}{\tan k}}}} \cdot \frac{1}{k}\right) \cdot \left(\left(\left(\frac{1}{k} \cdot \ell\right) \cdot \frac{1}{t}\right) \cdot \sqrt[3]{2 \cdot \frac{\cos k}{\sin k}}\right)\]
