$\frac{1}{ab}=\frac{s}{a}+\frac{r}{b} \overset{?}{\iff}\gcd(a,b)=1$ $$\frac{1}{ab}=\frac{s}{a}+\frac{r}{b} \overset{?}{\iff} \gcd(a,b)=1$$
This seems almost painfully obvious because it is just $ar+bs=1$ in another form. This second form is the definition of coprimality, so what else is my professor looking for?

If $\gcd(a,b)=1$ then since the greatest common divisor is the smallest positive integer that can be represented as a linear combination of a and b then we have that there are integers r and s such that 
$1=ra+sb$
By dividing by ab we have that 
$\frac{1}{ab}=\frac{s}{a}+\frac{r}{b}$.
Now if we suppose that $\frac{1}{ab}=\frac{s}{a}+\frac{r}{b}$ then by multiplying by ab we have 
$1=ra+sb$. Since $\gcd(a,b)$ divides both a and b then it divides 1 and since the greatest common divisor is non-negative then $\gcd(a,b)=1$.
