Initial program 32.4
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Initial simplification24.9
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
- Using strategy
rm Applied *-un-lft-identity24.9
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{1 \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Applied times-frac18.4
\[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{1 \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
Applied times-frac16.8
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{1} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Simplified16.8
\[\leadsto \color{blue}{\frac{\frac{2}{t}}{\sin k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
- Using strategy
rm Applied add-sqr-sqrt16.9
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\]
Applied *-un-lft-identity16.9
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Applied times-frac14.3
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Applied times-frac13.1
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{t}}{1}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\right)}\]
Applied associate-*r*11.7
\[\leadsto \color{blue}{\left(\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\]
- Using strategy
rm Applied add-cube-cbrt11.8
\[\leadsto \left(\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{\sqrt{\color{blue}{\left(\sqrt[3]{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*} \cdot \sqrt[3]{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right) \cdot \sqrt[3]{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Applied sqrt-prod11.8
\[\leadsto \left(\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{\color{blue}{\sqrt{\sqrt[3]{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*} \cdot \sqrt[3]{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}} \cdot \sqrt{\sqrt[3]{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Simplified11.8
\[\leadsto \left(\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{\color{blue}{\left|\sqrt[3]{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right|} \cdot \sqrt{\sqrt[3]{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Final simplification11.8
\[\leadsto \left(\frac{\frac{\ell}{t}}{\sqrt{\sqrt[3]{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}} \cdot \left|\sqrt[3]{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right|} \cdot \frac{\frac{2}{t}}{\sin k}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
