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
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t}}\]
- Using strategy
rm Applied times-frac31.1
\[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
Applied times-frac20.4
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{k}{t}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]
Simplified19.7
\[\leadsto \color{blue}{\frac{\frac{2}{k}}{\sin k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\]
- Using strategy
rm Applied *-un-lft-identity19.7
\[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{1 \cdot \frac{k}{t}}}\]
Applied *-un-lft-identity19.7
\[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{1 \cdot \frac{k}{t}}\]
Applied times-frac18.9
\[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{1 \cdot \frac{k}{t}}\]
Applied times-frac12.8
\[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{t}}{1}}{1} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\right)}\]
Applied associate-*r*11.2
\[\leadsto \color{blue}{\left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]
Simplified7.2
\[\leadsto \left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right) \cdot \color{blue}{\frac{\frac{\ell}{k}}{\tan k}}\]
- Using strategy
rm Applied frac-times6.5
\[\leadsto \color{blue}{\frac{\frac{2}{k} \cdot \frac{\frac{\ell}{t}}{1}}{\sin k \cdot 1}} \cdot \frac{\frac{\ell}{k}}{\tan k}\]
Applied associate-*l/6.7
\[\leadsto \color{blue}{\frac{\left(\frac{2}{k} \cdot \frac{\frac{\ell}{t}}{1}\right) \cdot \frac{\frac{\ell}{k}}{\tan k}}{\sin k \cdot 1}}\]
Simplified1.4
\[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{\frac{\tan k}{\frac{\ell}{k}}}}}{\sin k \cdot 1}\]
- Using strategy
rm Applied *-un-lft-identity1.4
\[\leadsto \frac{\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{\color{blue}{1 \cdot \frac{\tan k}{\frac{\ell}{k}}}}}{\sin k \cdot 1}\]
Applied times-frac1.0
\[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{k}}{1} \cdot \frac{\frac{2}{t}}{\frac{\tan k}{\frac{\ell}{k}}}}}{\sin k \cdot 1}\]
Applied associate-/l*1.0
\[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k}}{1}}{\frac{\sin k \cdot 1}{\frac{\frac{2}{t}}{\frac{\tan k}{\frac{\ell}{k}}}}}}\]
Simplified1.0
\[\leadsto \frac{\color{blue}{\frac{\ell}{k}}}{\frac{\sin k \cdot 1}{\frac{\frac{2}{t}}{\frac{\tan k}{\frac{\ell}{k}}}}}\]
Final simplification1.0
\[\leadsto \frac{\frac{\ell}{k}}{\frac{\sin k}{\frac{\frac{2}{t}}{\frac{\tan k}{\frac{\ell}{k}}}}}\]
