C. ##y= sinx##

We need to look for asymptotes here. Whenever there are asymptotes the domain will have restrictions.

A:

##y= cotx## can be written as ##y = cosx/sinx## by the quotient identity. There are vertical asymptotes whenever the denominator equals ##0## so if:

##sinx = 0##

Then

##x = 0 pi##

These will be the asymptotes in ##0 x < 2pi##. Therefore ##y =cotx## is not defined in all the real numbers.
B:
##y = secx## can be written as ##y = 1/cosx##. Vertical asymptotes in ##0 x < 2pi## will be at:
##cosx =0##
##x = pi/2 (3pi)/2##
Therefore ##y = secx## does not have a domain of all the real numbers.
C:
##y = sinx##
This has a denominator of ##1## or will never have a vertical asymptote. It is also continuous so this is the function we're looking for.
D:
##y = tanx## can be written as ##y = sinx/cosx## which will have asymptotes at ##x = pi/2## and ##x= (3pi)/2## in 0 x <2pi. It does not have a domain of all real numbers.
Hopefully this helps!