By the definition,  sec(x) = .


Thus the domain of your function is the set of all real numbers, where 

   cos(x) > 0.


Notice the sign "strictly greater than zero". 


The sets of real numbers, where cos(x) > 0, is the union of open intervals 


    U (,),  over all integer  k = 0, +/-1, +/-2, . . . 


The capital letter U, which you see the first in that line, denotes the Union of listed intervals.