If a and b had a nontrivial common factor k >= 2, then a = k*a' and b = k*b', so (ad - bc) = k(a'd - b'c) = ±1.
Alternatively, you must essentially show that a and b are coprime; that is the numerator and denominator share no common factor. Another way of saying this is to say that .
Let . We can write as . Thus must be either -1 or 1, and thus a and b are coprime.