When finding domains one looks for things that are undefined in the set of Real numbers. Among the things to avoid are:
Zeros in denominators
Negative radicands of even-numbered roots. For example:
Zero or negative arguments to logarithm functions
Your expression has two of these:
A denominator. It is hidden in the tan function! Since we cannot allow x values which make cos(x) be zero!
cos(x) = 0 when x = , , , etc. So we must exclude all these values from the domain.
An even-numbered root: square root. So we cannot allow tan(x) to be negative.
tan(x) < 0 when , , , .etc.
The domain is rest of the Real numbers: , , , , etc. Expressing this succinctly and completely is not easy. The domain is:
{ } where n is any integer.
Think about various integers. Subsitute them in for n above and see if you can recognize that you get one of the intervals listed or suggested by the "etc." list above.
The correct answer is
The domain of the given function is the union of all semi-intervals <= x < , for all integer n.
Each semi-interval includes its left endpoint and excludes its right endpoint.