Initial program 17.0
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
Initial simplification17.0
\[\leadsto R \cdot \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\]
- Using strategy
rm Applied cos-diff3.5
\[\leadsto R \cdot \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)} + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\]
- Using strategy
rm Applied log1p-expm1-u3.5
\[\leadsto R \cdot \color{blue}{\log_* (1 + (e^{\cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)} - 1)^*)}\]
- Using strategy
rm Applied add-exp-log3.5
\[\leadsto R \cdot \log_* (1 + (e^{\color{blue}{e^{\log \left(\cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\right)}}} - 1)^*)\]
- Using strategy
rm Applied acos-asin3.6
\[\leadsto R \cdot \log_* (1 + (e^{e^{\log \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\right)}}} - 1)^*)\]
Final simplification3.6
\[\leadsto \log_* (1 + (e^{e^{\log \left(\frac{\pi}{2} - \sin^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\right)}} - 1)^*) \cdot R\]
