Initial program 46.9
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
Simplified30.1
\[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{\left(\tan k \cdot \frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right) \cdot \sin k}}\]
- Using strategy
rm Applied associate-*r/30.6
\[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{k}{t} \cdot k}{t}}}}{\left(\tan k \cdot \frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right) \cdot \sin k}\]
Applied associate-/r/30.6
\[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t} \cdot k} \cdot t}}{\left(\tan k \cdot \frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right) \cdot \sin k}\]
Applied times-frac27.4
\[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{t} \cdot k}}{\tan k \cdot \frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}} \cdot \frac{t}{\sin k}}\]
Simplified21.4
\[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot k} \cdot \left(1 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)\right)}{k}} \cdot \frac{t}{\sin k}\]
- Using strategy
rm Applied associate-*l/23.1
\[\leadsto \frac{\frac{2}{\tan k \cdot k} \cdot \left(1 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{t}}{t}}\right)}{k} \cdot \frac{t}{\sin k}\]
Applied associate-*r/23.1
\[\leadsto \frac{\frac{2}{\tan k \cdot k} \cdot \color{blue}{\frac{1 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{t}}}{k} \cdot \frac{t}{\sin k}\]
Applied associate-*r/20.4
\[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k \cdot k} \cdot \left(1 \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)}{t}}}{k} \cdot \frac{t}{\sin k}\]
Applied associate-/l/20.5
\[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot k} \cdot \left(1 \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)}{k \cdot t}} \cdot \frac{t}{\sin k}\]
Simplified14.5
\[\leadsto \frac{\color{blue}{\left(\ell \cdot \frac{2}{\tan k}\right) \cdot \frac{\ell}{k \cdot t}}}{k \cdot t} \cdot \frac{t}{\sin k}\]
- Using strategy
rm Applied div-inv14.5
\[\leadsto \color{blue}{\left(\left(\left(\ell \cdot \frac{2}{\tan k}\right) \cdot \frac{\ell}{k \cdot t}\right) \cdot \frac{1}{k \cdot t}\right)} \cdot \frac{t}{\sin k}\]
Applied associate-*l*12.1
\[\leadsto \color{blue}{\left(\left(\ell \cdot \frac{2}{\tan k}\right) \cdot \frac{\ell}{k \cdot t}\right) \cdot \left(\frac{1}{k \cdot t} \cdot \frac{t}{\sin k}\right)}\]
- Using strategy
rm Applied associate-*r/12.0
\[\leadsto \left(\color{blue}{\frac{\ell \cdot 2}{\tan k}} \cdot \frac{\ell}{k \cdot t}\right) \cdot \left(\frac{1}{k \cdot t} \cdot \frac{t}{\sin k}\right)\]
Applied associate-*l/13.0
\[\leadsto \color{blue}{\frac{\left(\ell \cdot 2\right) \cdot \frac{\ell}{k \cdot t}}{\tan k}} \cdot \left(\frac{1}{k \cdot t} \cdot \frac{t}{\sin k}\right)\]
Applied associate-*l/13.0
\[\leadsto \color{blue}{\frac{\left(\left(\ell \cdot 2\right) \cdot \frac{\ell}{k \cdot t}\right) \cdot \left(\frac{1}{k \cdot t} \cdot \frac{t}{\sin k}\right)}{\tan k}}\]
Simplified1.3
\[\leadsto \frac{\color{blue}{2 \cdot \left(\frac{\frac{\frac{\ell}{k}}{t} \cdot t}{\sin k} \cdot \frac{\frac{\ell}{k}}{t}\right)}}{\tan k}\]
Final simplification1.3
\[\leadsto \frac{\left(\frac{t \cdot \frac{\frac{\ell}{k}}{t}}{\sin k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot 2}{\tan k}\]
