Initial program 32.2
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Initial simplification25.0
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
- Using strategy
rm Applied add-sqr-sqrt25.0
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]
Applied times-frac18.5
\[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}\]
Applied times-frac16.5
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]
- Using strategy
rm Applied add-sqr-sqrt16.5
\[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\color{blue}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}}\]
Applied sqrt-prod16.4
\[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}}\]
Applied *-un-lft-identity16.4
\[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]
Applied times-frac13.6
\[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]
Applied times-frac12.7
\[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{t}}{1}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)}\]
Simplified12.7
\[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \left(\color{blue}{\frac{\frac{\ell}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)\]
Final simplification12.7
\[\leadsto \left(\frac{\frac{\ell}{t}}{\sqrt{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}}\right) \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}\]
