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Moeen,
Could you share the PDF file link please where you found the circuit?
It's all in the wikipedia link.
Hello Moen,
I fear, I didn't exactly understand the problem. The low-pass based all-pass filter has a phase of -2arctan(ωRC), as calculated in your present link, and the high-pass based all-pass filter has a phase of 180°-2arctan(ωRC), as written in the Wikipedia article.
Best wishes,
Frank