In part (ii)(a), exp(-ΘM^1/3) / (1 - exp(-ΘM^1/3)) becomes 1 / (exp(ΘM^1/3) -1) I am not able to do this mathematically
It's been a long time since I did CT6 but I think you just have to divide the numerator and denominator by exp(-ΘM^1/3). This is allowed because you are essentially then multiplying by exp(-ΘM^1/3) /exp(-ΘM^1/3) i.e. multiplying by 1.