Are these equiv? $\cot^{-1}(-x) = -\cot^{-1}(x)$

Question

$-\cot(y)=x$

Let's say you want to put this in terms of y...

$\cot(y)=-x$

$y = \cot^{-1}(-x)$

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Is this also valid ?

$-\cot(y)=x$

$y = -\cot^{-1}(x)$

If yes, how exactly does one arrive that that result?

In other words, are these equivalent?

$\cot^{-1}(-x) = -\cot^{-1}(x)$

Answer 1

No and yes.

Recall that when you take the inverse, you need to take the inverse on both sides. So what you've actually done is $$-cot(y)=x$$ $$cot^{-1}(-cot(y))=cot^{-1}(x)$$ which is not equivalent to $y = -cot^{-1}(x)$.

However, it is true that $cot^{-1}(x)$ is an odd function.

Answer 2

I get this... So they differ by $\pi$...

So, the answer depends on your definition of $\mathrm{arccot}$.

added
According to Wikipedia we can define $y = \mathrm{arccot}(x)$ to mean: $x = \cot y$ and $0 < y < \pi$. The advantage is that it is a continuous function. And $\mathrm{arccot}\; x = \frac{\pi}{2} - \arctan x$