Maximising difference between the parameter and angle at centre for an ellipse

[sandmannz123]

I have the following question with a solution that I am having trouble understanding the very last stage. Hoping someone can help me understand the last part of the solution that I have highlighted.

Here is the question:



Here is the solution I have been given and I can understand all of this except the last part I have highlighted.

 

[Dr.Peterson]

I think that has to be a typo, and they meant to repeat what theta is. Phi can't equal theta when their difference is maximized!

Of course, there are actually four solutions, not just two they show; they've ignored the second and third quadrants.