Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 13E / a / iii / Solution

Every path can change sheets only by applying the two reflectionsTheir products satisfyThus an even number of reflections gives a translation by an integral multiple of $2\pi$, while an odd number gives a translated reflection. All possible values are thereforein agreement with the branches of the complex inverse sine.