Initial program 0.9
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
Initial simplification0.9
\[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*} + \lambda_1\]
- Using strategy
rm Applied sub-neg0.9
\[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*} + \lambda_1\]
Applied sin-sum0.9
\[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \cos \left(-\lambda_2\right) + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*} + \lambda_1\]
Simplified0.9
\[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\color{blue}{\sin \lambda_1 \cdot \cos \lambda_2} + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*} + \lambda_1\]
- Using strategy
rm Applied sub-neg0.9
\[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{(\left(\cos \phi_2\right) \cdot \left(\cos \color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right) + \left(\cos \phi_1\right))_*} + \lambda_1\]
Applied cos-sum0.2
\[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{(\left(\cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} + \left(\cos \phi_1\right))_*} + \lambda_1\]
Simplified0.2
\[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{(\left(\cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) + \left(\cos \phi_1\right))_*} + \lambda_1\]
- Using strategy
rm Applied add-log-exp0.3
\[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\color{blue}{\log \left(e^{(\left(\cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) + \left(\cos \phi_1\right))_*}\right)}} + \lambda_1\]
- Using strategy
rm Applied fma-udef0.3
\[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\log \left(e^{\color{blue}{\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) + \cos \phi_1}}\right)} + \lambda_1\]
Applied exp-sum0.3
\[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\log \color{blue}{\left(e^{\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} \cdot e^{\cos \phi_1}\right)}} + \lambda_1\]
Applied log-prod0.3
\[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\color{blue}{\log \left(e^{\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}\right) + \log \left(e^{\cos \phi_1}\right)}} + \lambda_1\]
Simplified0.2
\[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\color{blue}{(\left(\sin \lambda_2\right) \cdot \left(\sin \lambda_1\right) + \left(\cos \lambda_1 \cdot \cos \lambda_2\right))_* \cdot \cos \phi_2} + \log \left(e^{\cos \phi_1}\right)} + \lambda_1\]
Final simplification0.2
\[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \sin \left(-\lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\cos \phi_2 \cdot (\left(\sin \lambda_2\right) \cdot \left(\sin \lambda_1\right) + \left(\cos \lambda_2 \cdot \cos \lambda_1\right))_* + \log \left(e^{\cos \phi_1}\right)} + \lambda_1\]
