Why $x(t) = \cos(\frac{\pi}{3}t)$ and $y(t) = \sin(\frac{\pi}{3}t)$ are considered identical signals?
There is obviously an error in the problem formulation. From the given solution, the problem should read


  With $x(t) = \cos(\omega_x(t + \tau_x) + \theta_x)$ and
  $y(t) = \cos(\omega_y(t + \tau_y)+ \theta_y)$, determine for which of the
  following combinations $x(t)$ and $y(t)$ are identically equal for all
  $t$.


In this case the given solution $\omega_x=\omega_y$ and $\omega_x\tau_x+\theta_x=\omega_y\tau_y+\theta_y+2\pi k$ makes sense. The solution to the problem as it is given (with $y(t)=\sin(\omega_y(t + \tau_y)+ \theta_y)$), would be

$$\omega_x\tau_x+\theta_x=\omega_y\tau_y+\theta_y-\frac{\pi}{2}+2\pi k$$