EDIT: It appears as though the text has made a typo. The angle should be $\theta - \phi$. !!
How is angle $\angle OBW_2$ calculated in terms of $\theta$ or $\phi$ if the only angle measures given are $\phi$ , $90 - \theta$ , and $2\theta$.
The solution shows angle $\angle OBW_2$ as equal to $\theta$. How?
Figure:
There must be some mistakes in your supplied drawing.
Draw the red line through the center O parallel to $BW_2$. (I think the arrow lines are vertical.) Then, $\theta = x$ and $y = 2\theta – x = \theta$. In that case, the red line should be perpendicular to chord AB. If this is so, AB has to be horizontal.