Well, you keep asking me questions and my answers aren't working here, so let's reverse the process. You can answer the following.
What does a phasor represent?
What frequency is represented by a phasor?
What assumptions (such as linearity) are specified for the use of phasors?
The example in question, is it a linear or nonlinear operation?
You quoted an example, did you notice the answer they gave without the use of phasors? That is, did you see the DC component and sine wave with twice the frequency?
What frequency is implied in your answer, and does it match the correct answer?
Once you answer these questions, we can identify what the stumbling block is to understanding this issue. Note that one can always apply a method, even when the required assumptions are not true. Then you will get an incorrect answer. So, how do you know when an answer is correct? In this case, the answer is easy. Phasors give a fast way to get answers; so, do it the hard way and do it the phasor way and see if they match. If they don't match, why don't they match? The usual answer is "nonlinearity". Linear systems generally produce output signalss that are at the same frequency as the input signals. Nonlinear systems, generate additional frequencies.