Initial program 16.9
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
- Using strategy
rm Applied sub-neg16.9
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right) \cdot R\]
Applied cos-sum3.8
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}\right) \cdot R\]
Applied simplify3.8
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot R\]
- Using strategy
rm Applied sub-neg3.8
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \left(-\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)}\right) \cdot R\]
Applied distribute-lft-in3.8
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)}\right) \cdot R\]
Applied simplify3.8
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \color{blue}{\left(\cos \phi_1 \cdot \sin \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)}\right)\right) \cdot R\]
- Using strategy
rm Applied acos-asin3.8
\[\leadsto \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \sin \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)\right)} \cdot R\]
Applied simplify3.8
\[\leadsto \left(\frac{\pi}{2} - \color{blue}{\sin^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left((\left(\sin \lambda_2\right) \cdot \left(\sin \lambda_1\right) + \left(\cos \lambda_2 \cdot \cos \lambda_1\right))_*\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)}\right) \cdot R\]
- Using strategy
rm Applied add-log-exp3.9
\[\leadsto \left(\frac{\pi}{2} - \color{blue}{\log \left(e^{\sin^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left((\left(\sin \lambda_2\right) \cdot \left(\sin \lambda_1\right) + \left(\cos \lambda_2 \cdot \cos \lambda_1\right))_*\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)}\right)}\right) \cdot R\]
Applied add-log-exp3.9
\[\leadsto \left(\color{blue}{\log \left(e^{\frac{\pi}{2}}\right)} - \log \left(e^{\sin^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left((\left(\sin \lambda_2\right) \cdot \left(\sin \lambda_1\right) + \left(\cos \lambda_2 \cdot \cos \lambda_1\right))_*\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)}\right)\right) \cdot R\]
Applied diff-log3.9
\[\leadsto \color{blue}{\log \left(\frac{e^{\frac{\pi}{2}}}{e^{\sin^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left((\left(\sin \lambda_2\right) \cdot \left(\sin \lambda_1\right) + \left(\cos \lambda_2 \cdot \cos \lambda_1\right))_*\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)}}\right)} \cdot R\]
Applied simplify3.8
\[\leadsto \log \color{blue}{\left(e^{\frac{\pi}{2} - \sin^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left((\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right))_*\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)}\right)} \cdot R\]
