Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 316 / 1 / vi / Solution

Let $\Delta =\Omega -{\varphi }_d$. From part (iii), $\varphi -{\varphi }_d=\Delta +(\varphi -\Omega )$ and $\tan (\varphi -\Omega )=\cos I\tan f$. The tangent addition formula therefore givesor, without singular coordinate tangents,For $I=\pi /2-I^{\prime }$ with $I^{\prime }\ll 1$, smooth choice of the angular branch gives