Graphs of sec, csc and cot

$\sec\left(x\right) = \frac{1}{\cos\left(x\right)}$. Where the cosine is 1 or −1 the secant is the same; where the cosine shrinks towards 0 the secant grows without limit; where the cosine is 0 there is an asymptote. The graph is a row of U shapes, opening up above the cosine's tops and down below its bottoms, and it never takes a value between −1 and 1.

$\csc\left(x\right) = \frac{1}{\sin\left(x\right)}$ does the same to the sine: U shapes at its tops and bottoms, asymptotes where the sine is 0, at every multiple of π. Both have period 2π.

$\cot\left(x\right) = \frac{\cos\left(x\right)}{\sin\left(x\right)}$ has its asymptotes where the sine is 0, at multiples of π, its zeros where the cosine is 0, at π/2 + kπ, and falling branches with period π.