I assume that what you want to prove is
if gcd(a,b)=1 then gcd(a+b,ab)=1
One method of proof is the following:
- Assume that some integer greater than 1 divides a+b and ab, then there is a prime p that divides a+b and ab.
- Since p is a prime and p divides a and b it divided either a or b.
Assume p divides a.
- Show that since p divided a and a+b it also divided b.
- Thus gcd(a,b) ≥ p > 1