What is the range of $ y = (\operatorname{arccot} x) (\operatorname{ arccot} ( - x)) $

Question

What is the range of $ y = (\operatorname{ arccot x }) (\operatorname{ arccot{ - x }}) $. I solved this problem with right answer using AM GM inequality. But I received a lot of criticism for using AM GM inequality here on this site as it does not give sharp bounds. So is there a better way? I was thinking about Jensen's inequality but that doesn't work.

What is wrong with my solution of maximum value of $ \sin \frac {A}{2} + \sin \frac{B}{2} + \sin \frac{C}{2} $ in a triangle ABC?

The side of a triangle inscribed in a given circle subtends angles $a, b, $ and $ y$ at the center.

What is wrong with this solution of find the least value of $ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$

Answer 1

Like How do I prove that $\arccos(x) + \arccos(-x)=\pi$ when $x \in [-1,1]$?,

arccot$(x)\cdot$arccot$(-x)=$arccot$(x)(\pi-$arccot$(x))=\left(\dfrac\pi2\right)^2-\left(\text{arccot }(x)-\dfrac\pi2\right)^2$