There's a formula: another = an + other. Think of it as of an article plus the word "other" that have historically merged into one word. Grammar requires some article before "other book"; either "the" or "a." Depending on the context, you get either "You need to buy the other book" (if, for instance, the guy bought only the first book out of the set of two) or "You need to buy an_other book ...
A good example for this is a cube with 6 sides. And that is also the reason why you should use 'on the other hand' but 'on another side'. But I indicate that 'on the other side' and 'on the other hand' are just the same due to the use of 'other' here.
The string an other is vanishingly rare in English. In contrast another is positively pervasive. I think it would be fair to say that the second has eclipsed the first to the point of making the first unacceptable, even though it is a grammatical string. Both an and another are members of the category of determiners, while other, on the other hand, is an adjective. There's no grammatical ...
I have seen a statement: We have a Christian duty to serve each other. Is this statement correct? Because 'each other' is used for two persons and 'one another' for more than two persons.
pronouns - Is 'each other' used correctly when talking about sets of ...
The formal discourse marker used by the OP; "on the one hand ... on the other hand" expresses two contrasting ideas, opinions or facts. In view of this fact, I suggest the following alternatives: