You are solving the equation [MATH]\cos(4\theta) = 0[/MATH]. The angle you desire is not the only solution of that equation, so there are more than one solution. In fact, there are four solutions! So you have to use additional information to determine which is correct.
What has been used here is the fact that the solution is positive (so they didn't take the negative root), together with the stated form of the answer (that the coefficient of the radical is to be +1). But the latter is not really a sufficient reason, as it doesn't show that that form is correct. What should really be done is to determine which solution is in the proper interval for this cosine.