tan x has a vertical asymptote at x =

You just said to tell when a vertical asymptote occurs, so I will just tell one place that it occurs at for each of these. But keep in mind that if you add π, or 180º, another asymptote will occur there too. And each of these functions will have an infinite number of asymptotes because they are cyclic, which just means they repeat themselves over and over.

 

tan is sin/cos, and cos = 0 at π/2

tan x has a vertical asymptote at x = π/2

 

cot is cos/sin, and sin = 0 at 0π

cot x has a vertical asymptote at x = 0

 

sec is 1/cos, and cos = 0 at π/2

sec x has a vertical asymptote at x = π/2

 

csc is 1/sin, and sin = 0 at 0π

csc x has a vertical asymptote at x = 0

 

I put the answers in radians just because I think that makes the most sense for this question,

but if you need it in degrees: π/2 = 90º and 0 = 0º