Initial program 47.4
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
Initial simplification30.4
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t}}\]
- Using strategy
rm Applied times-frac30.2
\[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
Applied times-frac19.8
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{k}{t}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]
Simplified19.3
\[\leadsto \color{blue}{\frac{\frac{2}{k}}{\sin k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\]
- Using strategy
rm Applied *-un-lft-identity19.3
\[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{1 \cdot \frac{k}{t}}}\]
Applied add-cube-cbrt19.5
\[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}\right) \cdot \sqrt[3]{\tan k}}}}{1 \cdot \frac{k}{t}}\]
Applied times-frac18.7
\[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}}{1 \cdot \frac{k}{t}}\]
Applied times-frac12.8
\[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}{1} \cdot \frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}{\frac{k}{t}}\right)}\]
Applied associate-*r*11.6
\[\leadsto \color{blue}{\left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}{1}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}{\frac{k}{t}}}\]
Simplified7.4
\[\leadsto \left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}{1}\right) \cdot \color{blue}{\frac{\frac{\ell}{k}}{\sqrt[3]{\tan k}}}\]
- Using strategy
rm Applied *-un-lft-identity7.4
\[\leadsto \left(\frac{\frac{2}{k}}{\color{blue}{1 \cdot \sin k}} \cdot \frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}{1}\right) \cdot \frac{\frac{\ell}{k}}{\sqrt[3]{\tan k}}\]
Applied add-cube-cbrt7.5
\[\leadsto \left(\frac{\color{blue}{\left(\sqrt[3]{\frac{2}{k}} \cdot \sqrt[3]{\frac{2}{k}}\right) \cdot \sqrt[3]{\frac{2}{k}}}}{1 \cdot \sin k} \cdot \frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}{1}\right) \cdot \frac{\frac{\ell}{k}}{\sqrt[3]{\tan k}}\]
Applied times-frac7.5
\[\leadsto \left(\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{k}} \cdot \sqrt[3]{\frac{2}{k}}}{1} \cdot \frac{\sqrt[3]{\frac{2}{k}}}{\sin k}\right)} \cdot \frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}{1}\right) \cdot \frac{\frac{\ell}{k}}{\sqrt[3]{\tan k}}\]
Applied associate-*l*7.0
\[\leadsto \color{blue}{\left(\frac{\sqrt[3]{\frac{2}{k}} \cdot \sqrt[3]{\frac{2}{k}}}{1} \cdot \left(\frac{\sqrt[3]{\frac{2}{k}}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}{1}\right)\right)} \cdot \frac{\frac{\ell}{k}}{\sqrt[3]{\tan k}}\]
Simplified7.0
\[\leadsto \left(\color{blue}{\left(\sqrt[3]{\frac{2}{k}} \cdot \sqrt[3]{\frac{2}{k}}\right)} \cdot \left(\frac{\sqrt[3]{\frac{2}{k}}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}{1}\right)\right) \cdot \frac{\frac{\ell}{k}}{\sqrt[3]{\tan k}}\]
- Using strategy
rm Applied cbrt-div7.0
\[\leadsto \left(\left(\color{blue}{\frac{\sqrt[3]{2}}{\sqrt[3]{k}}} \cdot \sqrt[3]{\frac{2}{k}}\right) \cdot \left(\frac{\sqrt[3]{\frac{2}{k}}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}{1}\right)\right) \cdot \frac{\frac{\ell}{k}}{\sqrt[3]{\tan k}}\]
Applied associate-*l/7.0
\[\leadsto \left(\color{blue}{\frac{\sqrt[3]{2} \cdot \sqrt[3]{\frac{2}{k}}}{\sqrt[3]{k}}} \cdot \left(\frac{\sqrt[3]{\frac{2}{k}}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}{1}\right)\right) \cdot \frac{\frac{\ell}{k}}{\sqrt[3]{\tan k}}\]
Applied associate-*l/7.0
\[\leadsto \color{blue}{\frac{\left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{2}{k}}\right) \cdot \left(\frac{\sqrt[3]{\frac{2}{k}}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}{1}\right)}{\sqrt[3]{k}}} \cdot \frac{\frac{\ell}{k}}{\sqrt[3]{\tan k}}\]
Simplified1.8
\[\leadsto \frac{\color{blue}{\frac{\left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{2}{k}}\right) \cdot \frac{\sqrt[3]{\frac{2}{k}}}{\frac{\sin k}{\ell}}}{t \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}\right)}}}{\sqrt[3]{k}} \cdot \frac{\frac{\ell}{k}}{\sqrt[3]{\tan k}}\]
Final simplification1.8
\[\leadsto \frac{\frac{\ell}{k}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\frac{\sqrt[3]{\frac{2}{k}}}{\frac{\sin k}{\ell}} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{2}{k}}\right)}{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}\right) \cdot t}}{\sqrt[3]{k}}\]
