cot(x) = cos(x)/sin(x) = 1/tan(x)?

[Alex K]

Ok, another issue related to division by 0, this time in the context of trigonometry.

Since cot(x) is, as far as I know, the same as cos(x)/sin(x), then:

cot(pi/2) = cos(pi/2) / sin(pi/2) = 0/1 = 0 in radians

But cot(x) has also been equated to 1/tan(x) from what I've read, but if that's the case, then:

cot(pi/2) = 1/tan(pi/2) = 1/undefined = undefined?

So there's a contradiction here. In the case of pi/2, cos(x)/sin(x) != 1/tan(x). What gives? Is the answer found in calculus itself (with the limits and all)? Or is it a bit misleading to say that cot(x) = 1/tan(x)?

You're absolutely right in principle. At the points where cos is zero, this way of writing cot doesn't work. Often, one still writes the shorthand cot = 1/tan, and at isolated points where that is undefined, but the limit exists (for example for x -> pi/2) one takes it to mean the limit, which is

lim_(x->pi/2) 1/tan(x) = 0

But strictly speaking 1/tan doesn't work there.