Initial program 16.6
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
- Using strategy
rm Applied cos-diff3.7
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R\]
Applied distribute-rgt-in3.7
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)}\right) \cdot R\]
- Using strategy
rm Applied associate-*r*3.7
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2} + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right) \cdot R\]
- Using strategy
rm Applied add-exp-log3.7
\[\leadsto \color{blue}{e^{\log \left(\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2 + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)}} \cdot R\]
Final simplification3.7
\[\leadsto R \cdot e^{\log \left(\cos^{-1} \left(\left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right) + \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right) + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\]
