Initial program 16.9
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
Initial simplification16.9
\[\leadsto R \cdot \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\]
- Using strategy
rm Applied cos-diff4.0
\[\leadsto R \cdot \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)} + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\]
- Using strategy
rm Applied acos-asin4.1
\[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\right)}\]
- Using strategy
rm Applied asin-acos4.0
\[\leadsto R \cdot \left(\frac{\pi}{2} - \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\right)}\right)\]
Applied associate--r-4.0
\[\leadsto R \cdot \color{blue}{\left(\left(\frac{\pi}{2} - \frac{\pi}{2}\right) + \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\right)}\]
Applied distribute-lft-in4.0
\[\leadsto \color{blue}{R \cdot \left(\frac{\pi}{2} - \frac{\pi}{2}\right) + R \cdot \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)}\]
Simplified4.0
\[\leadsto R \cdot \left(\frac{\pi}{2} - \frac{\pi}{2}\right) + \color{blue}{R \cdot \cos^{-1} \left((\left((\left(\sin \lambda_2\right) \cdot \left(\sin \lambda_1\right) + \left(\cos \lambda_2 \cdot \cos \lambda_1\right))_*\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)}\]
Final simplification4.0
\[\leadsto \left(\frac{\pi}{2} - \frac{\pi}{2}\right) \cdot R + \cos^{-1} \left((\left((\left(\sin \lambda_2\right) \cdot \left(\sin \lambda_1\right) + \left(\cos \lambda_2 \cdot \cos \lambda_1\right))_*\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right) \cdot R\]
