Initial program 32.3
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
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
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \sin k}}{1} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Simplified16.0
\[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}}} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
- Using strategy
rm Applied div-inv16.0
\[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{t}}}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
Applied times-frac14.3
\[\leadsto \color{blue}{\left(\frac{2}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\frac{1}{t}}{\frac{\sin k}{\frac{\ell}{t}}}\right)} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
Applied associate-*l*12.8
\[\leadsto \color{blue}{\frac{2}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(\frac{\frac{1}{t}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right)}\]
- Using strategy
rm Applied associate-/r/13.2
\[\leadsto \frac{2}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(\color{blue}{\left(\frac{\frac{1}{t}}{\sin k} \cdot \frac{\ell}{t}\right)} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right)\]
Applied associate-*l*11.7
\[\leadsto \frac{2}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \color{blue}{\left(\frac{\frac{1}{t}}{\sin k} \cdot \left(\frac{\ell}{t} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right)\right)}\]
- Using strategy
rm Applied pow111.7
\[\leadsto \frac{2}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(\frac{\frac{1}{t}}{\sin k} \cdot \color{blue}{{\left(\frac{\ell}{t} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right)}^{1}}\right)\]
Applied pow111.7
\[\leadsto \frac{2}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(\color{blue}{{\left(\frac{\frac{1}{t}}{\sin k}\right)}^{1}} \cdot {\left(\frac{\ell}{t} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right)}^{1}\right)\]
Applied pow-prod-down11.7
\[\leadsto \frac{2}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \color{blue}{{\left(\frac{\frac{1}{t}}{\sin k} \cdot \left(\frac{\ell}{t} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right)\right)}^{1}}\]
Applied pow111.7
\[\leadsto \color{blue}{{\left(\frac{2}{\frac{\sin k}{\frac{\ell}{t}}}\right)}^{1}} \cdot {\left(\frac{\frac{1}{t}}{\sin k} \cdot \left(\frac{\ell}{t} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right)\right)}^{1}\]
Applied pow-prod-down11.7
\[\leadsto \color{blue}{{\left(\frac{2}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(\frac{\frac{1}{t}}{\sin k} \cdot \left(\frac{\ell}{t} \cdot \frac{\cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right)\right)\right)}^{1}}\]
Simplified11.6
\[\leadsto {\color{blue}{\left(\frac{\frac{\frac{\frac{\ell}{t} \cdot \cos k}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}{t \cdot \sin k}}{\frac{\frac{\sin k}{\frac{\ell}{t}}}{2}}\right)}}^{1}\]
Final simplification11.6
\[\leadsto \frac{\frac{\frac{\cos k \cdot \frac{\ell}{t}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}{\sin k \cdot t}}{\frac{\frac{\sin k}{\frac{\ell}{t}}}{2}}\]
