# Simplifying a trig expression

How can I get Mathematica to simplify the following expression?

Pi-ArcCos[-1/Sqrt]


I'd like the answer to be ArcCos[1/Sqrt], which seems manifestly "simpler" to me. If I use Simplify, it leaves it as it is. A related question is: howcome

Pi-ArcCos[-1/Sqrt]==ArcCos[1/Sqrt]


does not return True or False, but rather returns back the same expression?

• FullSimplify[Pi - ArcCos[-1/Sqrt] - ArcCos[1/Sqrt]] performs 0. Mar 17, 2021 at 18:33

Clear["Global*"]

expr = Pi - ArcCos[-1/Sqrt];


The simplification is available in MathematicalFunctionData

argSimp = (MathematicalFunctionData["ArcCos",
"ArgumentSimplifications"][][z] // Activate) /. Equal :> Rule

(* ArcCos[-z] -> π - ArcCos[z] *)

expr /. (argSimp /. z :> 1/Sqrt)

(* ArcCos[1/Sqrt] *)


EDIT: Use ComplexExpand to test the equality

Pi - ArcCos[-1/Sqrt] == ArcCos[1/Sqrt] // ComplexExpand

(* True *)


I would claim that your expected form is not the simplest one. I would instead propose $$\tan ^{-1}\left(\sqrt{2}\right)$$ and do the simplification as follows

e = Pi - ArcCos[-1/Sqrt];
es = ArcTan[Tan[e]]
FullSimplify[e==es]
(* ArcTan[Sqrt] *)
(* True *)
`