2007 AMC 12A Problem 17

Suppose that \sin a + \sin b = \sqrt{\frac{5}{3}} and \cos a + \cos b = 1. What is \cos (a - b)?

\mathrm{(A)}\ \sqrt{\frac{5}{3}} - 1\qquad \mathrm{(B)}\ \frac 13\qquad \mathrm{(C)}\ \frac 12\qquad \mathrm{(D)}\ \frac 23\qquad \mathrm{(E)}\ 1

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.