sp-acj4-205-cond true

category simplipy-conditional

Summary

Given

\(x \in \mathbb{C},\quad x \in i\mathbb{R},\quad \cos x \neq 0\)

Claim

\(\displaystyle \cosh\left(\operatorname{arsinh}\left(\tan x\right)\right) = \left|\frac{1}{\cos x}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[205] + rules.json[140]: cosh asinh tan _0 -> abs inv cos _0 [weakest assumption found: imaginary]

Lean proof: included

Answers (latest run per system; open code for the reproducer that ran)

system outcome answer steps (from the trace) s run query (as sent) detail

none

always-none no verdict null 0.0 2026-10-13 17:21:18 ba9dc4def822 (imaginary x) ∧ (nonzero (cos x)) ⊢ (= (cosh (asinh (tan x))) (abs (/ (cos x)))) null

sympy

sympy-ask no verdict null difference → sqrt(tan(x)**2 + 1) - 1/Abs(cos(x)) · ask → None 0.268 2026-10-13 17:21:19 ba9dc4def822 Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cos(x))) ⊢ Eq(cosh(asinh(tan(x))), Abs((1/cos(x)))) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), ~Q.zero(cos(x)))
Residual = cosh(asinh(tan(x))) - Abs(1 / cos(x))
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')

difference	`cosh(asinh(tan(x))) - Abs(1 / cos(x))`	→	`sqrt(tan(x)**2 + 1) - 1/Abs(cos(x))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → sqrt(tan(x)**2 + 1) - 1/Abs(cos(x)) · ask → None 0.045 2026-10-12 21:54:02 ee702ae2fe06 Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cos(x))) ⊢ Eq(cosh(asinh(tan(x))), Abs((1/cos(x)))) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), ~Q.zero(cos(x)))
Residual = cosh(asinh(tan(x))) - Abs(1 / cos(x))
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')

difference	`cosh(asinh(tan(x))) - Abs(1 / cos(x))`	→	`sqrt(tan(x)**2 + 1) - 1/Abs(cos(x))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine no verdict null difference → sqrt(tan(x)**2 + 1) - 1/Abs(cos(x)) · refine → sqrt(tan(x)**2 + 1) - 1/Abs(cos(x)) 0.076 2026-10-13 17:22:30 ba9dc4def822 Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cos(x))) ⊢ Eq(cosh(asinh(tan(x))), Abs((1/cos(x)))) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = refine(Abs(W), Q.positive(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), ~Q.zero(cos(x)))
Residual = cosh(asinh(tan(x))) - Abs(1 / cos(x))
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`cosh(asinh(tan(x))) - Abs(1 / cos(x))`	→	`sqrt(tan(x)**2 + 1) - 1/Abs(cos(x))`
refine	`refine(Residual, Assum)`	→	`sqrt(tan(x)**2 + 1) - 1/Abs(cos(x))`
answer			`null`

sympy-refine no verdict null difference → sqrt(tan(x)**2 + 1) - 1/Abs(cos(x)) · refine → sqrt(tan(x)**2 + 1) - 1/Abs(cos(x)) 0.064 2026-10-12 21:54:16 ee702ae2fe06 Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cos(x))) ⊢ Eq(cosh(asinh(tan(x))), Abs((1/cos(x)))) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = refine(Abs(W), Q.positive(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), ~Q.zero(cos(x)))
Residual = cosh(asinh(tan(x))) - Abs(1 / cos(x))
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`cosh(asinh(tan(x))) - Abs(1 / cos(x))`	→	`sqrt(tan(x)**2 + 1) - 1/Abs(cos(x))`
refine	`refine(Residual, Assum)`	→	`sqrt(tan(x)**2 + 1) - 1/Abs(cos(x))`
answer			`null`

sympy-simplify solved true difference → sqrt(tan(x)**2 + 1) - 1/Abs(cos(x)) · simplify → 0 0.117 2026-10-13 17:24:01 ba9dc4def822 Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cos(x))) ⊢ Eq(cosh(asinh(tan(x))), Abs((1/cos(x)))) true

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True, imaginary=True)
# dropped (not a unary fact about one symbol): ~Q.zero(cos(x))
Residual = cosh(asinh(tan(x))) - Abs(1 / cos(x))
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`cosh(asinh(tan(x))) - Abs(1 / cos(x))`	→	`sqrt(tan(x)**2 + 1) - 1/Abs(cos(x))`
simplify	`simplify(Residual)`	→	`0`
answer			`true`

sympy-simplify solved true difference → sqrt(tan(x)**2 + 1) - 1/Abs(cos(x)) · simplify → 0 0.112 2026-10-12 21:54:43 ee702ae2fe06 Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cos(x))) ⊢ Eq(cosh(asinh(tan(x))), Abs((1/cos(x)))) true

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True, imaginary=True)
# dropped (not a unary fact about one symbol): ~Q.zero(cos(x))
Residual = cosh(asinh(tan(x))) - Abs(1 / cos(x))
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`cosh(asinh(tan(x))) - Abs(1 / cos(x))`	→	`sqrt(tan(x)**2 + 1) - 1/Abs(cos(x))`
simplify	`simplify(Residual)`	→	`0`
answer			`true`

wolfram

wolfram-fullsimplify no verdict null difference → -Abs[Sec[x]] + Sqrt[1 + Tan[x]^2] · result → -Abs[Sec[x]] + Sqrt[Sec[x]^2] 0.271 2026-10-13 17:24:42 ba9dc4def822 Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cos[x] != 0) ⊢ (Cosh[ArcSinh[Tan[x]]] == Abs[(1/Cos[x])]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cos[x] != 0));
  Residual = (Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]])`	→	`-Abs[Sec[x]] + Sqrt[1 + Tan[x]^2]`
result	`FullSimplify[Residual, Assum]`	→	`-Abs[Sec[x]] + Sqrt[Sec[x]^2]`
answer			`null`

wolfram-refine no verdict null difference → -Abs[Sec[x]] + Sqrt[1 + Tan[x]^2] · result → -Abs[Sec[x]] + Sqrt[1 + Tan[x]^2] 0.022 2026-10-13 17:24:31 ba9dc4def822 Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cos[x] != 0) ⊢ (Cosh[ArcSinh[Tan[x]]] == Abs[(1/Cos[x])]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cos[x] != 0));
  Residual = (Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]])`	→	`-Abs[Sec[x]] + Sqrt[1 + Tan[x]^2]`
result	`Refine[Residual, Assum]`	→	`-Abs[Sec[x]] + Sqrt[1 + Tan[x]^2]`
answer			`null`

wolfram-simplify no verdict null difference → -Abs[Sec[x]] + Sqrt[1 + Tan[x]^2] · result → -Abs[Sec[x]] + Sqrt[Sec[x]^2] 0.081 2026-10-13 17:24:17 ba9dc4def822 Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cos[x] != 0) ⊢ (Cosh[ArcSinh[Tan[x]]] == Abs[(1/Cos[x])]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cos[x] != 0));
  Residual = (Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]])`	→	`-Abs[Sec[x]] + Sqrt[1 + Tan[x]^2]`
result	`Simplify[Residual, Assum]`	→	`-Abs[Sec[x]] + Sqrt[Sec[x]^2]`
answer			`null`

Older runs (11)

system outcome answer steps (from the trace) s run query (as sent) detail

wolfram-fullsimplify no verdict null difference → -Abs[Sec[x]] + Sqrt[1 + Tan[x]^2] · result → -Abs[Sec[x]] + Sqrt[Sec[x]^2] 0.273 2026-10-13 16:43:00 3ba0dfac4bca Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cos[x] != 0) ⊢ (Cosh[ArcSinh[Tan[x]]] == Abs[(1/Cos[x])]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cos[x] != 0));
  Residual = (Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]])`	→	`-Abs[Sec[x]] + Sqrt[1 + Tan[x]^2]`
result	`FullSimplify[Residual, Assum]`	→	`-Abs[Sec[x]] + Sqrt[Sec[x]^2]`
answer			`null`

wolfram-refine no verdict null difference → -Abs[Sec[x]] + Sqrt[1 + Tan[x]^2] · result → -Abs[Sec[x]] + Sqrt[1 + Tan[x]^2] 0.022 2026-10-13 16:42:50 3ba0dfac4bca Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cos[x] != 0) ⊢ (Cosh[ArcSinh[Tan[x]]] == Abs[(1/Cos[x])]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cos[x] != 0));
  Residual = (Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]])`	→	`-Abs[Sec[x]] + Sqrt[1 + Tan[x]^2]`
result	`Refine[Residual, Assum]`	→	`-Abs[Sec[x]] + Sqrt[1 + Tan[x]^2]`
answer			`null`

wolfram-simplify no verdict null difference → -Abs[Sec[x]] + Sqrt[1 + Tan[x]^2] · result → -Abs[Sec[x]] + Sqrt[Sec[x]^2] 0.081 2026-10-13 16:42:35 3ba0dfac4bca Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cos[x] != 0) ⊢ (Cosh[ArcSinh[Tan[x]]] == Abs[(1/Cos[x])]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cos[x] != 0));
  Residual = (Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]])`	→	`-Abs[Sec[x]] + Sqrt[1 + Tan[x]^2]`
result	`Simplify[Residual, Assum]`	→	`-Abs[Sec[x]] + Sqrt[Sec[x]^2]`
answer			`null`

sympy-simplify solved true difference → sqrt(tan(x)**2 + 1) - 1/Abs(cos(x)) · simplify → 0 0.12 2026-10-13 16:42:18 3ba0dfac4bca Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cos(x))) ⊢ Eq(cosh(asinh(tan(x))), Abs((1/cos(x)))) true

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True, imaginary=True)
# dropped (not a unary fact about one symbol): ~Q.zero(cos(x))
Residual = cosh(asinh(tan(x))) - Abs(1 / cos(x))
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`cosh(asinh(tan(x))) - Abs(1 / cos(x))`	→	`sqrt(tan(x)**2 + 1) - 1/Abs(cos(x))`
simplify	`simplify(Residual)`	→	`0`
answer			`true`

sympy-refine no verdict null difference → sqrt(tan(x)**2 + 1) - 1/Abs(cos(x)) · refine → sqrt(tan(x)**2 + 1) - 1/Abs(cos(x)) 0.077 2026-10-13 16:40:40 3ba0dfac4bca Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cos(x))) ⊢ Eq(cosh(asinh(tan(x))), Abs((1/cos(x)))) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = refine(Abs(W), Q.positive(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), ~Q.zero(cos(x)))
Residual = cosh(asinh(tan(x))) - Abs(1 / cos(x))
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`cosh(asinh(tan(x))) - Abs(1 / cos(x))`	→	`sqrt(tan(x)**2 + 1) - 1/Abs(cos(x))`
refine	`refine(Residual, Assum)`	→	`sqrt(tan(x)**2 + 1) - 1/Abs(cos(x))`
answer			`null`

always-none no verdict null 0.0 2026-10-13 16:39:28 3ba0dfac4bca (imaginary x) ∧ (nonzero (cos x)) ⊢ (= (cosh (asinh (tan x))) (abs (/ (cos x)))) null

sympy-ask no verdict null difference → sqrt(tan(x)**2 + 1) - 1/Abs(cos(x)) · ask → None 0.269 2026-10-13 16:39:28 3ba0dfac4bca Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cos(x))) ⊢ Eq(cosh(asinh(tan(x))), Abs((1/cos(x)))) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), ~Q.zero(cos(x)))
Residual = cosh(asinh(tan(x))) - Abs(1 / cos(x))
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')

difference	`cosh(asinh(tan(x))) - Abs(1 / cos(x))`	→	`sqrt(tan(x)**2 + 1) - 1/Abs(cos(x))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify no verdict null difference → -Abs[Sec[x]] + Sqrt[1 + Tan[x]^2] · result → -Abs[Sec[x]] + Sqrt[Sec[x]^2] 0.272 2026-10-12 21:55:24 ee702ae2fe06 Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cos[x] != 0) ⊢ (Cosh[ArcSinh[Tan[x]]] == Abs[(1/Cos[x])]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cos[x] != 0));
  Residual = (Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]])`	→	`-Abs[Sec[x]] + Sqrt[1 + Tan[x]^2]`
result	`FullSimplify[Residual, Assum]`	→	`-Abs[Sec[x]] + Sqrt[Sec[x]^2]`
answer			`null`

wolfram-refine no verdict null difference → -Abs[Sec[x]] + Sqrt[1 + Tan[x]^2] · result → -Abs[Sec[x]] + Sqrt[1 + Tan[x]^2] 0.022 2026-10-12 21:55:14 ee702ae2fe06 Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cos[x] != 0) ⊢ (Cosh[ArcSinh[Tan[x]]] == Abs[(1/Cos[x])]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cos[x] != 0));
  Residual = (Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]])`	→	`-Abs[Sec[x]] + Sqrt[1 + Tan[x]^2]`
result	`Refine[Residual, Assum]`	→	`-Abs[Sec[x]] + Sqrt[1 + Tan[x]^2]`
answer			`null`

always-none no verdict null 0.0 2026-10-12 21:55:00 ee702ae2fe06 (imaginary x) ∧ (nonzero (cos x)) ⊢ (= (cosh (asinh (tan x))) (abs (/ (cos x)))) null

wolfram-simplify no verdict null difference → -Abs[Sec[x]] + Sqrt[1 + Tan[x]^2] · result → -Abs[Sec[x]] + Sqrt[Sec[x]^2] 0.081 2026-10-12 21:55:00 ee702ae2fe06 Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cos[x] != 0) ⊢ (Cosh[ArcSinh[Tan[x]]] == Abs[(1/Cos[x])]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cos[x] != 0));
  Residual = (Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Cosh[ArcSinh[Tan[x]]]) - (Abs[1/Cos[x]])`	→	`-Abs[Sec[x]] + Sqrt[1 + Tan[x]^2]`
result	`Simplify[Residual, Assum]`	→	`-Abs[Sec[x]] + Sqrt[Sec[x]^2]`
answer			`null`

Source (conditional.sexp)

(case sp-acj4-205-cond (category simplipy-conditional)
  (note "SimpliPy acj-4 (MIT) rules_real.json[205] + rules.json[140]: cosh asinh tan _0  ->  abs inv cos _0  [weakest assumption found: imaginary]")
  (var x complex)
  (assume (imaginary x))
  (assume (nonzero (cos x)))
  (ask (= (cosh (asinh (tan x))) (abs (/ (cos x)))))
  (expect true)
  (lean-proof "obtain ⟨y, hy, rfl⟩ := AE.cond_imaginary_exists h0.2; refine ⟨⟨⟨⟨(AE.tan.D_iff _).mpr (by simpa using h1.2), by simp⟩, by simp⟩, ⟨⟨by simp, h1.2⟩, by simp⟩⟩, ?_⟩; rw [AE.tan_eq, AE.cond_tan_mul_I', AE.cond_asinh_mul_I_of (AE.cond_tanh_mem _).1 (AE.cond_tanh_mem _).2, AE.cosh_eq, AE.cond_cosh_mul_I', Real.cos_arcsin, AE.cond_one_sub_tanh_sq, Real.sqrt_inv, Real.sqrt_sq (Real.cosh_pos y).le, AE.cos_eq, AE.cond_cos_mul_I', AE.inv_def, ← Complex.ofReal_inv, AE.abs_ofReal, abs_of_pos (inv_pos.mpr (Real.cosh_pos y))]"))