If author used square brackets inside #cos# just as brackets (not as an absolute value, which it looks like in a small font), than here is the explanation.
Also it should be noted that the square root in this problem used without #+-# sign implies arithmetic square root, that is only its non-negative value.
Square root requires #cos x# to be non-negative, which entails that #x# must be in the intervals #[-pi/2+2pi n,pi/2+2pi n]# for all integer #n#. That assures non-negativity of #cos x# and, therefore, constitutes the domain of this function.
In this domain #cos x# ranges in the interval #[0,1]# and square root of it also will be in the same range.