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-frac30.3
\[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
Applied times-frac19.6
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{k}{t}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]
Simplified19.0
\[\leadsto \color{blue}{\frac{\frac{2}{k}}{\sin k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\]
- Using strategy
rm Applied *-un-lft-identity19.0
\[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{1 \cdot \frac{k}{t}}}\]
Applied *-un-lft-identity19.0
\[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{1 \cdot \frac{k}{t}}\]
Applied times-frac18.3
\[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{1 \cdot \frac{k}{t}}\]
Applied times-frac12.5
\[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{t}}{1}}{1} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\right)}\]
Applied associate-*r*11.0
\[\leadsto \color{blue}{\left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]
Simplified6.9
\[\leadsto \left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right) \cdot \color{blue}{\frac{\frac{\ell}{k}}{\tan k}}\]
- Using strategy
rm Applied associate-*l/6.4
\[\leadsto \color{blue}{\frac{\frac{2}{k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}}{\sin k}} \cdot \frac{\frac{\ell}{k}}{\tan k}\]
Simplified1.0
\[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{2}{t}}}{\sin k} \cdot \frac{\frac{\ell}{k}}{\tan k}\]
- Using strategy
rm Applied div-inv1.0
\[\leadsto \frac{\frac{\ell}{k} \cdot \frac{2}{t}}{\sin k} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{1}{\tan k}\right)}\]
Applied associate-*r*1.3
\[\leadsto \color{blue}{\left(\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{\sin k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\tan k}}\]
Final simplification1.3
\[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k} \cdot \frac{2}{t}}{\sin k}\right) \cdot \frac{1}{\tan k}\]
