I have just done part (iii) of this question and can get the right answer but am a bit confused why do we take arcosh i.e. just the principle value of cosh and not the other value. I presume this is the case with trig? also in this question they have ignored the negative square root when using a trigh identity please explain why?? (a=.5, b-0.25)
It is admittedly traditional to write the substitution as "Let $2x=\cosh u$." But for completeness it should have been "Let $2x=\cosh u$, where $u\ge 0$." (Or else, conceivably, where $u\le 0$.)
Then $u=\text{arccosh}(2x)$ is the only possibility, the alternative is not available. Or else, more directly, we could have described the substitution as $u=\text{arccosh}(2x)$, in which case there is no ambiguity of sign.
Whenever we make a substitution that involves a function which is not one to one, we must restrict the domain.