AMC 12 2007a Problem 17

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

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