The terminal ray of the standard-position angle 270° intersects the unit circle at (0, -1) = (cos(270°), sin(270°)). That is, cos(270°) = 0.
The secant is the reciprocal of the cosine, so the value of sec(270°) is
sec(270°) = 1/cos(270°) = 1/0, which is undefined.
_____
The relationships between trigonometric functions are determined by their definitions. Reference to the unit circle proves nothing about their relationships.
