Initial program 47.0
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
- Using strategy
rm Applied add-sqr-sqrt47.0
\[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\sqrt{1 + {\left(\frac{k}{t}\right)}^{2}} \cdot \sqrt{1 + {\left(\frac{k}{t}\right)}^{2}}} - 1\right)}\]
Applied difference-of-sqr-147.0
\[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(\sqrt{1 + {\left(\frac{k}{t}\right)}^{2}} + 1\right) \cdot \left(\sqrt{1 + {\left(\frac{k}{t}\right)}^{2}} - 1\right)\right)}}\]
Applied associate-*r*47.0
\[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\sqrt{1 + {\left(\frac{k}{t}\right)}^{2}} + 1\right)\right) \cdot \left(\sqrt{1 + {\left(\frac{k}{t}\right)}^{2}} - 1\right)}}\]
Applied simplify39.8
\[\leadsto \frac{2}{\color{blue}{\left(\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}} \cdot (\left(\tan k\right) \cdot \left(\sqrt{1^2 + \left(\frac{k}{t}\right)^2}^*\right) + \left(\tan k\right))_*\right)} \cdot \left(\sqrt{1 + {\left(\frac{k}{t}\right)}^{2}} - 1\right)}\]
- Using strategy
rm Applied pow139.8
\[\leadsto \frac{2}{\left(\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}} \cdot (\left(\tan k\right) \cdot \left(\sqrt{1^2 + \left(\frac{k}{t}\right)^2}^*\right) + \left(\tan k\right))_*\right) \cdot \color{blue}{{\left(\sqrt{1 + {\left(\frac{k}{t}\right)}^{2}} - 1\right)}^{1}}}\]
Applied pow139.8
\[\leadsto \frac{2}{\left(\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}} \cdot \color{blue}{{\left((\left(\tan k\right) \cdot \left(\sqrt{1^2 + \left(\frac{k}{t}\right)^2}^*\right) + \left(\tan k\right))_*\right)}^{1}}\right) \cdot {\left(\sqrt{1 + {\left(\frac{k}{t}\right)}^{2}} - 1\right)}^{1}}\]
Applied pow139.8
\[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{1}} \cdot {\left((\left(\tan k\right) \cdot \left(\sqrt{1^2 + \left(\frac{k}{t}\right)^2}^*\right) + \left(\tan k\right))_*\right)}^{1}\right) \cdot {\left(\sqrt{1 + {\left(\frac{k}{t}\right)}^{2}} - 1\right)}^{1}}\]
Applied pow-prod-down39.8
\[\leadsto \frac{2}{\color{blue}{{\left(\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}} \cdot (\left(\tan k\right) \cdot \left(\sqrt{1^2 + \left(\frac{k}{t}\right)^2}^*\right) + \left(\tan k\right))_*\right)}^{1}} \cdot {\left(\sqrt{1 + {\left(\frac{k}{t}\right)}^{2}} - 1\right)}^{1}}\]
Applied pow-prod-down39.8
\[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}} \cdot (\left(\tan k\right) \cdot \left(\sqrt{1^2 + \left(\frac{k}{t}\right)^2}^*\right) + \left(\tan k\right))_*\right) \cdot \left(\sqrt{1 + {\left(\frac{k}{t}\right)}^{2}} - 1\right)\right)}^{1}}}\]
Applied simplify33.8
\[\leadsto \frac{2}{{\color{blue}{\left(\frac{(\left(\tan k\right) \cdot \left(\sqrt{1^2 + \left(\frac{k}{t}\right)^2}^*\right) + \left(\tan k\right))_*}{\frac{\frac{\ell}{t}}{t \cdot \sin k}} \cdot \frac{\sqrt{1^2 + \left(\frac{k}{t}\right)^2}^* - 1}{\frac{\ell}{t}}\right)}}^{1}}\]
Taylor expanded around inf 22.6
\[\leadsto \frac{2}{{\color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}}^{1}}\]
Applied simplify10.0
\[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{\frac{\sin k \cdot \sin k}{\frac{\cos k}{t}}}}\]
- Using strategy
rm Applied div-inv10.0
\[\leadsto \frac{\frac{2}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{\frac{\sin k \cdot \sin k}{\color{blue}{\cos k \cdot \frac{1}{t}}}}\]
Applied times-frac8.5
\[\leadsto \frac{\frac{2}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{\color{blue}{\frac{\sin k}{\cos k} \cdot \frac{\sin k}{\frac{1}{t}}}}\]
Applied *-un-lft-identity8.5
\[\leadsto \frac{\color{blue}{1 \cdot \frac{2}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}}{\frac{\sin k}{\cos k} \cdot \frac{\sin k}{\frac{1}{t}}}\]
Applied times-frac7.8
\[\leadsto \color{blue}{\frac{1}{\frac{\sin k}{\cos k}} \cdot \frac{\frac{2}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{\frac{\sin k}{\frac{1}{t}}}}\]
Applied simplify7.8
\[\leadsto \color{blue}{\frac{\cos k}{\sin k}} \cdot \frac{\frac{2}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{\frac{\sin k}{\frac{1}{t}}}\]
Applied simplify1.5
\[\leadsto \frac{\cos k}{\sin k} \cdot \color{blue}{\frac{\frac{\frac{2}{t}}{\frac{k}{\ell}}}{\frac{k}{\ell} \cdot \sin k}}\]
