Cos[sqrt[-x]] == Cosh[sqrt[x]] but not Cos[sqrt[-(x+y)]] == Cosh[sqrt[(x+y)]]?

I have trouble with the following apparent contradiction:

In:= FullSimplify[Cos[Sqrt[-a]] == Cosh[Sqrt[a]]]
Out= True

In:= FullSimplify[Cos[Sqrt[-(b + c)]] == Cosh[Sqrt[(b + c)]]]
Out= Cos[Sqrt[-b - c]] == Cosh[Sqrt[b + c]]

I also tried...

In:= FullSimplify[ Cos[Sqrt[-(b + c)]] == Cosh[Sqrt[(b + c)]] /. {b -> a - c}]
Out= True

In:= FullSimplify[Cos[Sqrt[-(b + c)]] == Cosh[Sqrt[(b + c)]], b \[Element] Reals && c \[Element] Reals]
Out= Cos[Sqrt[-b - c]] == Cosh[Sqrt[b + c]]

In:= FullSimplify[Cos[Sqrt[-(b + c)]] == Cosh[Sqrt[(b + c)]], b > 0 && c > 0]
Out= True

...but is there a way so resolve this without an explicit substitution or without making assumptions (which weren't necessary for a in the first place)?

(running Mathematica 12.0.0.0 on MS Windows (64-bit))

• g[Power[x_, 1/2]] := -I*Power[-x, 1/2]; Simplify[TrigToExp[Cos[Sqrt[-(b + c)]] == Cosh[Sqrt[(b + c)]]], TransformationFunctions -> {Automatic, g}]
– Acus
Nov 17 '21 at 7:40
• Works very fast, thank you! Is there a way to modify this for arguments in the form (Sqrt[-a^2 + b^2 + Sqrt[c]] d)/e or in general arbitrary arguments? Nov 17 '21 at 9:21
• Not quite clear what your mean. Added g[ ] rule is rather general (do not restrict structure of argument of Sqrt[ ], the FullForm of which is Power[_,1/2 ]). You can keep adding as many rules as your want on your own risk. Another very useful option to Simplify family is ComplexityFunction (read in doc).
– Acus
Nov 17 '21 at 9:44
• For example g[Power[x_, 1/2]] := -I*Power[-x, 1/2]; Simplify[TrigToExp[Cos[Sqrt[-a^2 + b^2 + Sqrt[c]] ] == Cosh[Sqrt[a^2 - b^2 - Sqrt[c]]]], TransformationFunctions -> {Automatic, g}] does not work. Nov 17 '21 at 10:32
• sqrtRule = Power[x_, 1/2] :> RuleCondition[-I*Power[-x, 1/2], LeafCount[x] > LeafCount[Expand[-x]]]; Cos[Sqrt[-a^2 + b^2 + Sqrt[c]]] == Cosh[Sqrt[a^2 - b^2 - Sqrt[c]]] /. sqrtRule
– Acus
Nov 17 '21 at 11:40

Edit ... (Contrary to first result, true without conditions)

Splitting b and c into real and imaginary part together with Reduce helps to show second equation beeing True.

red2 = Reduce[
b1 \[Element] Reals && b2 \[Element] Reals && c1 \[Element] Reals &&
c2 \[Element] Reals &&
Cos[Sqrt[-(b + c)]] == Cosh[Sqrt[(b + c)]] /. {b -> b1 + I b2,
c -> c1 + I c2}, {b1, b2, c1, c2}, GeneratedParameters -> d]

(*   ((b1 | b2 | c1 | c2) \[Element] Reals &&
b2 + c2 != 0) || ((b1 | b2) \[Element]
Reals && ((c1 > -b1 && c2 == -b2) || (c1 == -b1 &&
c2 == -b2) || (c1 < -b1 && c2 == -b2))) || (d \[Element]
Integers && (b1 | b2) \[Element]
Reals && ((d >= 1 && c1 == -b1 - 4 \[Pi]^2 d^2 &&
c2 == -b2) || (d >= 0 &&
c1 == -b1 - \[Pi]^2 - 4 \[Pi]^2 d - 4 \[Pi]^2 d^2 &&
c2 == -b2)))   *)

FullSimplify[red2,
b1 \[Element] Reals && b2 \[Element] Reals && c1 \[Element] Reals &&
c2 \[Element] Reals]

(*   True   *)

The same can be done for the first equation.

• The output of Reduce[Cos[Sqrt[-a]] == Cosh[Sqrt[a]], a, Complexes] is simply wrong. Cos[Sqrt[-a]] and Cosh[Sqrt[a]] are identical entire functions (see Order 1/2 in that Wiki article). Their power expansions are identical: execute Series[Cos[Sqrt[-a]] {a, 0, n}] and Series[ Cosh[Sqrt[a]], {a, 0, n}] with any n. Nov 16 '21 at 19:42
• Of course, positive integer. Nov 16 '21 at 20:01
• Thank you for this clarification! Using Reduce is a bit slow, though, so I guess I'll go with the series expansion suggested by user64494 and Acus Nov 17 '21 at 9:22