Initial program 31.3
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Initial simplification24.6
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
- Using strategy
rm Applied *-un-lft-identity24.6
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{1 \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Applied tan-quot24.6
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \color{blue}{\frac{\sin k}{\cos k}}}}{1 \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
Applied associate-*r/24.6
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\color{blue}{\frac{\sin k \cdot \sin k}{\cos k}}}}{1 \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
Applied associate-/r/24.6
\[\leadsto \frac{\color{blue}{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \sin k} \cdot \cos k}}{1 \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
Applied times-frac24.6
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \sin k}}{1} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Simplified16.3
\[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}}} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
- Using strategy
rm Applied *-un-lft-identity16.3
\[\leadsto \frac{\color{blue}{1 \cdot \frac{2}{t}}}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
Applied times-frac14.5
\[\leadsto \color{blue}{\left(\frac{1}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\frac{2}{t}}{\frac{\sin k}{\frac{\ell}{t}}}\right)} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
Applied associate-*l*12.8
\[\leadsto \color{blue}{\frac{1}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(\frac{\frac{2}{t}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right)}\]
Simplified12.8
\[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \left(\frac{\frac{2}{t}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right)\]
- Using strategy
rm Applied associate-*l/11.7
\[\leadsto \frac{\frac{\ell}{t}}{\sin k} \cdot \color{blue}{\frac{\frac{2}{t} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}{\frac{\sin k}{\frac{\ell}{t}}}}\]
Applied associate-*r/11.7
\[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{\sin k} \cdot \left(\frac{2}{t} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right)}{\frac{\sin k}{\frac{\ell}{t}}}}\]
- Using strategy
rm Applied frac-times11.7
\[\leadsto \frac{\frac{\frac{\ell}{t}}{\sin k} \cdot \color{blue}{\frac{2 \cdot \cos k}{t \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}{\frac{\sin k}{\frac{\ell}{t}}}\]
Applied associate-*r/11.7
\[\leadsto \frac{\color{blue}{\frac{\frac{\frac{\ell}{t}}{\sin k} \cdot \left(2 \cdot \cos k\right)}{t \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}{\frac{\sin k}{\frac{\ell}{t}}}\]
Applied associate-/l/11.7
\[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{\sin k} \cdot \left(2 \cdot \cos k\right)}{\frac{\sin k}{\frac{\ell}{t}} \cdot \left(t \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right)}}\]
Final simplification11.7
\[\leadsto \frac{\left(\cos k \cdot 2\right) \cdot \frac{\frac{\ell}{t}}{\sin k}}{\frac{\sin k}{\frac{\ell}{t}} \cdot \left(t \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right)}\]
