A visual approach is to consider what the graph looks like. An even function is symmetrical over the y-axis. y = CotX is not even.
An odd function has point symmetry about the origin (0,0). If you rotate the graph of y=CotX about (0,0), it matches the original graph. So it is an odd function.
The algebraic method is to consider that an even function satisfies the condition f(x) = f(-x). Simply substituting x = 30 deg into your calculator will show that 2Cot(30) = 3.464 while 2Cot(-30) = -3.464 which are not the same.
An odd function must satisfy f(-x) = -f(x), which the value for 30 deg clearly satisfies.
To prove that 2Cot(-x) = -2Cot(x), the condition for it to be an odd function, we use the trigonometric identities
Cot(x) =(Cos(x))/(Sin(x)
Cos(-x) = Cos(x)
Sin(-x) = - Sin(x)
Then,
2Cot(-x) = 2((cos(-x))/sin(-x))
2cot(-x) = 2((cos(x))/(-sin(x)))
2cot(-x) = -2((cos(x))/sin(x))
2cot(-x) = -2cot(x)
Q.E.D.