Initial program 32.1
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Initial simplification24.6
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}\]
- Using strategy
rm Applied add-sqr-sqrt24.7
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}}}\]
Applied times-frac18.5
\[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}}\]
Applied times-frac16.6
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}}}\]
- Using strategy
rm Applied *-un-lft-identity16.6
\[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}}\]
Applied times-frac13.9
\[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}}\]
Applied associate-/l*13.0
\[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}} \cdot \color{blue}{\frac{\frac{\frac{\ell}{t}}{1}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}}{\frac{\frac{\ell}{t}}{\tan k}}}}\]
Simplified13.0
\[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}} \cdot \frac{\color{blue}{\frac{\ell}{t}}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}}{\frac{\frac{\ell}{t}}{\tan k}}}\]
- Using strategy
rm Applied add-sqr-sqrt13.0
\[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\color{blue}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}}}} \cdot \frac{\frac{\ell}{t}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}}{\frac{\frac{\ell}{t}}{\tan k}}}\]
Applied sqrt-prod13.0
\[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}}}} \cdot \frac{\frac{\ell}{t}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}}{\frac{\frac{\ell}{t}}{\tan k}}}\]
Final simplification13.0
\[\leadsto \frac{\frac{\ell}{t}}{\frac{\sqrt{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}}}{\frac{\frac{\ell}{t}}{\tan k}}} \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\sqrt{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \sqrt{\sqrt{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}}}}\]
