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And I solved it the following way.

Assume one is y1 = A sin (wt) and the other one y2 = A sin (wt + phi)

max(y1-y2) = A (given)

Solve for phi.

Since both y1 and y2 are sinusoids of w frequency, their difference will also be a sinusoid with a different amplitude but with same frequency and a different phase.

Since we are just interested in the max value of the difference, I calculated the amplitude of the sinusoid using phasors A e^j(phi) - A. It turned out to be 2A sin(phi/2). From that I calculated phi as pi/3 but the answer I have is as pi/2. I think the book is correct but I could not find the mistake in my calculation.

Can some one point out if I am wrong and if yes, where?