Which function has a domain of all real numbers? A.y=cotx B.y=secx C. y=sinx D.y=tanx

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!