Fringe Patterns. See attached file for full problem description.
The solution is in the attached file 70476.pdf
There is some kind of a confusion in that there is only one $theta$ in the picture, but you mention $theta_1$ and $theta_2$ further down in the text.
I shall assume that what is meant in the text is that $theta_1= theta$ and
On how to find a period:
Suppose we have a function of x ...