You are right, and your book is wrong: not just a little bit wrong, or wrong under certain limited circumstances, but entirely wrong.
The sentence, as written, has an unambiguous meaning:
It is possible that he will get rid of his cough if he smokes less in the future.
The words in which this is expressed are unimpeachably correct in any register—this is not a ‘formal’ vs ‘colloquial’ issue, or anything of that sort. It does not require any ‘improvement’.
The sentence proposed as an ‘improvement’ is no such thing: it is exactly as grammatical, no more, no less, but it presents a different meaning:
It is certain that he would get rid of his cough if he smoked less now or in the future.
Unhappily, textbooks often present deplorable exercises like this. The authors are typically very narrowly focused on the particular grammatical “rule” they are currently teaching†; to test your understanding they invent sentences which seem to violate that rule, and they fail to realize that their sentences have a meaning beyond the context in which they invented them, and that the “violations” are in fact compliant under different rules.
You are to be congratulated on spotting this error. Don’t hold it against the author: the folks who write these textbooks are mostly overworked, underpaid teachers who haven’t got time to exercise critical imagination, proofread and rewrite three or four or a dozen times.
† I'm going to bet that this exercise follows either a section of the book on 'first/second/third conditionals' or a section of the book introducing the use of the modal verbs can/could, may/might, must, shall/should and will/would.