Final answer:
When the tangent graph has a vertical asymptote, the cotangent function is typically undefined or tends to infinity as it is reciprocal to the tangent.
Step-by-step explanation:
The correct answer is option when the tangent graph has a vertical asymptote, the cotangent function is equal to zero. Remember that the tangent function is defined as sin(x)/cos(x) and cotangent as cos(x)/sin(x). The vertical asymptotes of the tangent occur where cos(x) is zero because the function is not defined when we divide by zero.
At these same points, since the sine is not zero, the value of the cotangent function, which is the reciprocal of the tangent, would also be undefined or tend to infinity.
However, when asked specifically for the value of cotangent 'at' the vertical asymptote (an impossibility since the function is not defined there), the question may be interpreted improperly.
Perhaps it's meant to ask what happens to cotangent as it approaches its own zeroes, which correspond to the vertical asymptotes of the tangent graph; in this case, as x approaches the zeroes of sine (where tangent has vertical asymptotes), cotangent approaches infinity, or is undefined, not zero.
The vertical asymptotes of the tangent function occur at odd multiples of pi/2. When the tangent graph has a vertical asymptote, the cotangent function is equal to zero. This is because the cotangent of an angle is the reciprocal of the tangent of that angle.
For example, if the tangent graph has a vertical asymptote at pi/2, then the cotangent function will be zero when evaluated at pi/2.