Question

For what numbers x, -2π ≤ x ≤ 2π, does the graph of y = csc(x) have vertical asymptotes?

For what numbers x, -2π ≤ x ≤ 2π, does the graph of y = cot(x) have vertical asymptotes?

Answer 1

By definition csc(x) = 1/sin(x). Hence csc(x) = 1 when sin(x) = 1.

So when does sin(x) = 1 in that interval?

Answer 2

csc(x) has vertical asymptotes when sin(x)=0 since csc(x)=1/sin(x)
cot(x) has vertical asymptotes when sin(x)=0 since cot(x)=cos(x)/sin(x)