smc

One can square both sides to get $a^2+b^2=a^2+2ab+b^2$. Then, $0=2ab⟹ab=0$. Also, it is clear that both sides of the equation must be nonnegative. The answer is $\boxed{\text{It is true if and only if }ab=0\text{ and }a+b\ge 0\text{.}}$.