Initial program 31.6
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Simplified24.5
\[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\left(\sin k \cdot \tan k\right) \cdot \frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\]
- Using strategy
rm Applied tan-quot24.5
\[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\left(\sin k \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}\]
Applied associate-*r/24.5
\[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\color{blue}{\frac{\sin k \cdot \sin k}{\cos k}} \cdot \frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}\]
Applied associate-*l/24.5
\[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\color{blue}{\frac{\left(\sin k \cdot \sin k\right) \cdot \frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}{\cos k}}}\]
Simplified14.4
\[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\color{blue}{\left(t \cdot \frac{\sin k}{\frac{\ell}{t}}\right) \cdot \frac{\sin k}{\frac{\ell}{t}}}}{\cos k}}\]
- Using strategy
rm Applied *-un-lft-identity14.4
\[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\left(t \cdot \frac{\sin k}{\frac{\ell}{t}}\right) \cdot \frac{\sin k}{\frac{\ell}{t}}}{\color{blue}{1 \cdot \cos k}}}\]
Applied times-frac14.4
\[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\color{blue}{\frac{t \cdot \frac{\sin k}{\frac{\ell}{t}}}{1} \cdot \frac{\frac{\sin k}{\frac{\ell}{t}}}{\cos k}}}\]
Applied add-cube-cbrt14.4
\[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}\right) \cdot \sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}}}{\frac{t \cdot \frac{\sin k}{\frac{\ell}{t}}}{1} \cdot \frac{\frac{\sin k}{\frac{\ell}{t}}}{\cos k}}\]
Applied times-frac12.6
\[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{t \cdot \frac{\sin k}{\frac{\ell}{t}}}{1}} \cdot \frac{\sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{\frac{\sin k}{\frac{\ell}{t}}}{\cos k}}}\]
- Using strategy
rm Applied *-un-lft-identity12.6
\[\leadsto \frac{\sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{t \cdot \frac{\sin k}{\frac{\ell}{t}}}{\color{blue}{1 \cdot 1}}} \cdot \frac{\sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{\frac{\sin k}{\frac{\ell}{t}}}{\cos k}}\]
Applied times-frac12.6
\[\leadsto \frac{\sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\color{blue}{\frac{t}{1} \cdot \frac{\frac{\sin k}{\frac{\ell}{t}}}{1}}} \cdot \frac{\sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{\frac{\sin k}{\frac{\ell}{t}}}{\cos k}}\]
Applied associate-/r*11.5
\[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{t}{1}}}{\frac{\frac{\sin k}{\frac{\ell}{t}}}{1}}} \cdot \frac{\sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{\frac{\sin k}{\frac{\ell}{t}}}{\cos k}}\]
Simplified11.5
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{t}{1}}}{\color{blue}{\frac{\sin k}{\frac{\ell}{t}}}} \cdot \frac{\sqrt[3]{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{\frac{\sin k}{\frac{\ell}{t}}}{\cos k}}\]
Final simplification11.5
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \sqrt[3]{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}}}{t}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}}}{\frac{\frac{\sin k}{\frac{\ell}{t}}}{\cos k}}\]
