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

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 $\sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$

• Can you please provide some context and show what you have tried? What is more, there are some issues with your LaTeX code. Use \operatorname{arccot} instead of \arccot. May 14 '19 at 12:52
• @PantelisSopasakis I did say I solved this problem with right answer using AM GM what more could I add? May 14 '19 at 12:54
• @PantelisSopasakis anything else I should add?? May 14 '19 at 13:03
• @PantelisSopasakis questions are put on hold even if they follow "this guide"math.meta.stackexchange.com/questions/30088/…. So I doubt how helpful it is. May 14 '19 at 13:14
• You're right, MSE doesn't work perfectly. I meant to say that the better you phrase your questions, the higher the chances that you'll get an answer. May 14 '19 at 13:18

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