Initial program 31.4
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Initial simplification24.8
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
- Using strategy
rm Applied *-un-lft-identity24.8
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{1 \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}}\]
Applied tan-quot24.8
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \color{blue}{\frac{\sin k}{\cos k}}}}{1 \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]
Applied associate-*r/24.8
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\color{blue}{\frac{\sin k \cdot \sin k}{\cos k}}}}{1 \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]
Applied associate-/r/24.8
\[\leadsto \frac{\color{blue}{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \sin k} \cdot \cos k}}{1 \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]
Applied times-frac24.8
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \sin k}}{1} \cdot \frac{\cos k}{\frac{k}{t} \cdot \frac{k}{t} + 2}}\]
Simplified16.1
\[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}}} \cdot \frac{\cos k}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
- Using strategy
rm Applied *-un-lft-identity16.1
\[\leadsto \frac{\color{blue}{1 \cdot \frac{2}{t}}}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\cos k}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
Applied times-frac14.3
\[\leadsto \color{blue}{\left(\frac{1}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\frac{2}{t}}{\frac{\sin k}{\frac{\ell}{t}}}\right)} \cdot \frac{\cos k}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
Applied associate-*l*12.7
\[\leadsto \color{blue}{\frac{1}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(\frac{\frac{2}{t}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\cos k}{\frac{k}{t} \cdot \frac{k}{t} + 2}\right)}\]
Simplified12.7
\[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \left(\frac{\frac{2}{t}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\cos k}{\frac{k}{t} \cdot \frac{k}{t} + 2}\right)\]
- Using strategy
rm Applied associate-*l/11.5
\[\leadsto \frac{\frac{\ell}{t}}{\sin k} \cdot \color{blue}{\frac{\frac{2}{t} \cdot \frac{\cos k}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\sin k}{\frac{\ell}{t}}}}\]
Applied associate-*r/11.5
\[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{\sin k} \cdot \left(\frac{2}{t} \cdot \frac{\cos k}{\frac{k}{t} \cdot \frac{k}{t} + 2}\right)}{\frac{\sin k}{\frac{\ell}{t}}}}\]
- Using strategy
rm Applied frac-times11.6
\[\leadsto \frac{\frac{\frac{\ell}{t}}{\sin k} \cdot \color{blue}{\frac{2 \cdot \cos k}{t \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}}}{\frac{\sin k}{\frac{\ell}{t}}}\]
Applied associate-*r/11.6
\[\leadsto \frac{\color{blue}{\frac{\frac{\frac{\ell}{t}}{\sin k} \cdot \left(2 \cdot \cos k\right)}{t \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}}}{\frac{\sin k}{\frac{\ell}{t}}}\]
Simplified10.2
\[\leadsto \frac{\frac{\frac{\frac{\ell}{t}}{\sin k} \cdot \left(2 \cdot \cos k\right)}{\color{blue}{\frac{k \cdot k}{t} + 2 \cdot t}}}{\frac{\sin k}{\frac{\ell}{t}}}\]
Final simplification10.2
\[\leadsto \frac{\frac{\left(\cos k \cdot 2\right) \cdot \frac{\frac{\ell}{t}}{\sin k}}{2 \cdot t + \frac{k \cdot k}{t}}}{\frac{\sin k}{\frac{\ell}{t}}}\]
