Initial program 16.4
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
- Using strategy
rm Applied cos-diff3.6
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R\]
Applied distribute-lft-in3.6
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R\]
- Using strategy
rm Applied add-log-exp3.6
\[\leadsto \color{blue}{\log \left(e^{\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}\right)} \cdot R\]
Applied simplify3.6
\[\leadsto \log \color{blue}{\left(e^{\cos^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left((\left(\sin \lambda_2\right) \cdot \left(\sin \lambda_1\right) + \left(\cos \lambda_2 \cdot \cos \lambda_1\right))_*\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)}\right)} \cdot R\]
- Using strategy
rm Applied acos-asin3.7
\[\leadsto \log \left(e^{\color{blue}{\frac{\pi}{2} - \sin^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left((\left(\sin \lambda_2\right) \cdot \left(\sin \lambda_1\right) + \left(\cos \lambda_2 \cdot \cos \lambda_1\right))_*\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)}}\right) \cdot R\]
