Cotangent domain error?

Question

Some time ago I was wondering about the definition of cotangent; namely, that it is both defined as $\frac{\cos x}{\sin x}$ and as $\frac{1}{\tan x}$. However, $\cot(90°)$ and $\cot(270°)$ are equal to 0. Through the first definition I listed of cotangent, this is equal to $$\cot(90°) =\frac{\cos(90°)}{\sin(90°)} = \frac{0}{1} = 0$$

With the second definition, this is equal to $$\cot(90°) = \frac{1}{\tan(90°)} = \frac{1}{undefined} =\ ?$$

While I know that at $\tan(90°)$ there is an asymptote which, from the right goes to $-\infty$ and from the left goes to $\infty$. While I understand that in this case, the limit $\lim_{x\to \frac{\pi}{2}}\frac{1}{\tan x} = 0$, wouldn't this constitute a domain error in the function $\cot x$ since $\frac{1}{\infty}$ technically doesn't equal $0$?

Thanks for reading.

Answer 1

Cotangent is not $\frac 1\tan$. That is not the definition, and never has been. It just so happens that the two functions $\cot$ and $\frac1\tan$ agree on almost all of $\Bbb R$. When that happens, there is a strict sense in which we consider the functions equivalent, and some authors misuse the symbol $=$ to represent this.

That being said, kudos to you for stopping up and thinking about it.

Answer 2

Just $$\cot{x}\neq\frac{1}{\tan{x}}$$ on domain of $\cot.$

We can write $\cot{x}=\frac{1}{\tan{x}}$ for $\sin{x}\cos{x}\neq0$ only.