Initial program 31.9
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Initial simplification24.9
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
- Using strategy
rm Applied *-un-lft-identity24.9
\[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{1 \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}}\]
Applied times-frac18.7
\[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{1 \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]
Applied times-frac17.0
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{1} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 2}}\]
Simplified17.0
\[\leadsto \color{blue}{\frac{\frac{2}{t}}{\sin k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
- Using strategy
rm Applied add-sqr-sqrt17.1
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]
Applied *-un-lft-identity17.1
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}\]
Applied times-frac14.5
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}\]
Applied times-frac13.2
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{t}}{1}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}\right)}\]
Simplified13.2
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \left(\color{blue}{\frac{\frac{\ell}{t}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}\right)\]
- Using strategy
rm Applied add-sqr-sqrt13.2
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\color{blue}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}}\right)\]
Applied *-un-lft-identity13.2
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)\]
Applied *-un-lft-identity13.2
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\color{blue}{1 \cdot \frac{\ell}{t}}}{1 \cdot \tan k}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)\]
Applied times-frac13.2
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\color{blue}{\frac{1}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)\]
Applied times-frac13.2
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \color{blue}{\left(\frac{\frac{1}{1}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)}\right)\]
Simplified13.2
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \left(\color{blue}{\frac{1}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)\right)\]
Final simplification13.2
\[\leadsto \frac{\frac{2}{t}}{\sin k} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}} \cdot \frac{1}{\sqrt{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}}\right)\right)\]
