Can anyone explain those marked steps in case (iii)

follows from the fact that the cosine function is -periodic, which means . Roughly speaking, this is the same as saying that a point on a circle is the same as the point you get by completing a full revolution around the circle (i.e. add to the original point's angle with respect to the horizontal axis).

If you make another complete revolution (so we're effectively adding ) we get the same result: . This is true for any number of complete revolutions, so that this pattern holds for any even multiple of added to the argument. Therefore for any integer .

Next, because , it follows that is also true for any integer . So we have



The rest follows from considering either case and solving for .