Your answers are clearly wrong since they don't satisfy the original equations.
Your mistake is in forgetting that $\displaystyle \theta - \phi = \pi$ is only one of several possible solutions to $\displaystyle \cos (\theta - \phi) = -1$. Other solutions include $\displaystyle -\pi$ .....
When you solve
$\displaystyle \theta - \phi = - \pi$ .... (1)
$\displaystyle \theta + \phi = \frac{5 \pi}{3}$ .... (2)
simultaneously, you get the books answer (which is correct by the way).