I am querying the method. To solve for cos 2x =1 , 0<x<360 , we change the interval to 0<2x<540. That's what I originally meant to write. I shall post my workings in a few minutes when I get back to desk. I think in the original case one would divide 360 by 2 then add 45 degrees.
This is a very different approach than I use; so your question was not about the problem, but about your specific method, which you didn't state.
You never got back on this, so I'll comment on what you have said about it.
Using this method for the example here, the idea is not to find a new interval for x, but for the argument of the cosine. The interval 0<x<360 corresponds to 0<2x<720, by doubling each term. Why did you say 540? I may be missing part of what your method is, or you may just have typed wrong.
Now for the equation you were asking about,
3cos(x/2+45)=1 0<=x<=360 degrees
Here we want an interval containing x/2+45. So we can divide each term by 2, 0<=x/2<=360/2, and then add 45 degrees to each term:
0+45<=x/2+45<=360/2+45
That is,
45<=x/2+45<=225
So it appears that you did the right thing. You find all angles whose cosine is 1/3, between 45 and 225 degrees, and then solve for x in each case.