Does O(f(n)) + O(g(n)) = O(max{f(n), g(n)})?
One way to read this "identity" is as follows:


  If $a = O(f(n))$ and $b = O(g(n))$ then $a+b = O(\max\{f(n),g(n)\})$.


This indeed holds, as you can prove using the definition of big O.

There is another possible interpretation:


  $\{ a + b : a = O(f(n)), \; b = O(g(n)) \} = \{ c : c = O(\max\{f(n),g(n)\})\}$.


This also holds, as you van prove using the definition.

Whether it is a legitimate claim in an exam is up to your professor. It is definitely a legitimate claim in a paper.