What happens to the tangent function at ±90 degrees?

The tangent function, defined as the ratio of the sine to the cosine (tan(θ) = sin(θ)/cos(θ)), exhibits specific behavior at various angles. At ±90 degrees, the cosine of the angle equals zero (cos(90°) = 0 and cos(-90°) = 0). When we attempt to compute the tangent at these angles, we observe that division by zero occurs since tan(±90°) involves dividing by zero.

This results in the tangent function being undefined at ±90 degrees because it cannot yield a numerical value when the denominator is zero. Thus, recognizing that at these angles the tangent function does not exist explains why the correct answer is indeed that it is undefined.