Thanks for you time. I don't mean to say you are not right as it might well can be. On the first pass, I don't think that's what the book meant, the reason is, this example is right at the beginning of the chapter and the book did not mention anything about the concaveness of the star. Also the answer of the book:
Which does not contain any compensation for the concaveness of the planet. So far, this book has been pretty straight forward and concise. I type out the exact words from the book in my post and nothing about this. I scan through the later part of the chapter and there's nothing about this either. Basically it only talked about the ratio of the solid angle for object smaller than the half power angle.
Again thank you for your time. I manage to go on line and use a cosine calculator that don't flag error of 0.0025 deg and gave me 2 more digits in the decimal place than my calculator, the answer is even more off!! So I don't think the small angle is the cause of the problem.