Initial program 47.2
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
Initial simplification30.2
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}\]
- Using strategy
rm Applied add-sqr-sqrt30.2
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}}\]
Applied times-frac30.1
\[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}\]
Applied times-frac27.2
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}}\]
Simplified27.2
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}\]
Simplified13.0
\[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \color{blue}{\left(\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{\ell}{t}}{\left|\frac{k}{t}\right|}\right)}\]
- Using strategy
rm Applied associate-*r*11.8
\[\leadsto \color{blue}{\left(\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{\frac{\ell}{t}}{\left|\frac{k}{t}\right|}}\]
- Using strategy
rm Applied add-sqr-sqrt11.8
\[\leadsto \left(\frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{\sqrt{\left|\frac{k}{t}\right|} \cdot \sqrt{\left|\frac{k}{t}\right|}}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{\frac{\ell}{t}}{\left|\frac{k}{t}\right|}\]
Applied *-un-lft-identity11.8
\[\leadsto \left(\frac{\color{blue}{1 \cdot \frac{\frac{2}{t}}{\sin k}}}{\sqrt{\left|\frac{k}{t}\right|} \cdot \sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{\frac{\ell}{t}}{\left|\frac{k}{t}\right|}\]
Applied times-frac11.8
\[\leadsto \left(\color{blue}{\left(\frac{1}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\left|\frac{k}{t}\right|}}\right)} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{\frac{\ell}{t}}{\left|\frac{k}{t}\right|}\]
Final simplification11.8
\[\leadsto \frac{\frac{\ell}{t}}{\left|\frac{k}{t}\right|} \cdot \left(\left(\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{1}{\sqrt{\left|\frac{k}{t}\right|}}\right) \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\]
