To have vertical asymptotes, the denominator of a function must equal 0.
We know sin is 0 at Θ= 0, pi, 2pi, 3pi, 4pi........npi. And at both + and -.
Ok now let's look at the functions.
csc x = 1/sin x
cos x = cos x and has no denominator
tan x = sin x /cos x
cot x = cos x / sin x
The only places where denominator can be 0 is csc x and cot x so D is your answer.