1) ctg(5π/2):
Since cotangent is the reciprocal of tangent, we can find the cotangent of an angle by taking the reciprocal of the tangent of that angle.
The tangent of 5π/2 is undefined because it is at a vertical asymptote. Therefore, the cotangent of 5π/2 is also undefined.

2) ctg(-31π/6):
To find the cotangent of -31π/6, we first need to find the coterminal angle in the range 0 to 2π (0 to 12π/6).
-31π/6 + 12π = 1π/6
The cotangent of 1π/6 is √3. Therefore, cot(-31π/6) = √3.

3) ctg(4π/3):
The cotangent of an angle is the reciprocal of the tangent of that angle.
The tangent of 4π/3 is -√3, so the cotangent of 4π/3 is the reciprocal of -√3, which is -1/√3 or -√3/3.