← suite trig

trig-sqrt-01 contingent

category identities

Summary

Given
\(x \in \mathbb{R}\)
Claim
\(\displaystyle \sqrt{1 - \sin x \, \sin x} \overset{?}{=} \cos x\)
Expected
contingent true at \(x = 0\); false at \(x = 3\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 3\)

sqrt(cos^2) is |cos x|, which is cos x only where cos x >= 0.

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

systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
none
always-nonedeclined correctlynull0.02026-10-13 17:21:03 7a1dcfa636ed(real x) ⊢ (= (sqrt (- 1 (* (sin x) (sin x)))) (cos x))null (correct: contingent)
sympy
sympy-askdeclined correctlynulldifference → sqrt(1 - sin(x)**2) - cos(x) · ask → None0.4062026-10-13 17:21:03 7a1dcfa636edQ.real(x) ⊢ Eq(sqrt((1 - (sin(x)*sin(x)))), cos(x))null (correct: contingent)
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', real=True)
Assum = And(Q.real(x))
Residual = sqrt(1 - sin(x) * sin(x)) - 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')
differencesqrt(1 - sin(x) * sin(x)) - cos(x)→sqrt(1 - sin(x)**2) - cos(x)
askQ.zero(Residual)→None
answernull
sympy-askno verdictnulldifference → sqrt(1 - sin(x)**2) - cos(x) · ask → None0.0462026-10-12 20:21:44 45d6b9056f06Q.real(x) ⊢ Eq(sqrt((1 - (sin(x)*sin(x)))), 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', real=True)
Assum = And(Q.real(x))
Residual = sqrt(1 - sin(x) * sin(x)) - 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')
differencesqrt(1 - sin(x) * sin(x)) - cos(x)→sqrt(1 - sin(x)**2) - cos(x)
askQ.zero(Residual)→None
answernull
sympy-refinedeclined correctlynulldifference → sqrt(1 - sin(x)**2) - cos(x) · refine → sqrt(1 - sin(x)**2) - cos(x)0.0532026-10-13 17:21:06 7a1dcfa636edQ.real(x) ⊢ Eq(sqrt((1 - (sin(x)*sin(x)))), cos(x))null (correct: contingent)
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', real=True)
Assum = And(Q.real(x))
Residual = sqrt(1 - sin(x) * sin(x)) - 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')
differencesqrt(1 - sin(x) * sin(x)) - cos(x)→sqrt(1 - sin(x)**2) - cos(x)
refinerefine(Residual, Assum)→sqrt(1 - sin(x)**2) - cos(x)
answernull
sympy-refineno verdictnulldifference → sqrt(1 - sin(x)**2) - cos(x) · refine → sqrt(1 - sin(x)**2) - cos(x)0.0482026-10-12 20:21:45 45d6b9056f06Q.real(x) ⊢ Eq(sqrt((1 - (sin(x)*sin(x)))), 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', real=True)
Assum = And(Q.real(x))
Residual = sqrt(1 - sin(x) * sin(x)) - 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')
differencesqrt(1 - sin(x) * sin(x)) - cos(x)→sqrt(1 - sin(x)**2) - cos(x)
refinerefine(Residual, Assum)→sqrt(1 - sin(x)**2) - cos(x)
answernull
sympy-simplifydeclined correctlynulldifference → sqrt(1 - sin(x)**2) - cos(x) · simplify → -cos(x) + Abs(cos(x))0.0612026-10-13 17:21:07 7a1dcfa636edQ.real(x) ⊢ Eq(sqrt((1 - (sin(x)*sin(x)))), cos(x))null (correct: contingent)
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', real=True)
Residual = sqrt(1 - sin(x) * sin(x)) - 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')
differencesqrt(1 - sin(x) * sin(x)) - cos(x)→sqrt(1 - sin(x)**2) - cos(x)
simplifysimplify(Residual)→-cos(x) + Abs(cos(x))
answernull
sympy-simplifyno verdictnulldifference → sqrt(1 - sin(x)**2) - cos(x) · simplify → -cos(x) + Abs(cos(x))0.0632026-10-12 20:21:47 45d6b9056f06Q.real(x) ⊢ Eq(sqrt((1 - (sin(x)*sin(x)))), 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 = 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', real=True)
Residual = sqrt(1 - sin(x) * sin(x)) - 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')
differencesqrt(1 - sin(x) * sin(x)) - cos(x)→sqrt(1 - sin(x)**2) - cos(x)
simplifysimplify(Residual)→-cos(x) + Abs(cos(x))
answernull
wolfram
wolfram-fullsimplify declined correctlynulldifference → -Cos[x] + Sqrt[1 - Sin[x]^2] · result → Abs[Cos[x]] - Cos[x]0.0952026-10-13 17:21:15 7a1dcfa636edElement[x, Reals] ⊢ (Sqrt[(1 - (Sin[x]*Sin[x]))] == Cos[x])null (correct: contingent)
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, Reals]);
  Residual = (Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x])→-Cos[x] + Sqrt[1 - Sin[x]^2]
resultFullSimplify[Residual, Assum]→Abs[Cos[x]] - Cos[x]
answernull
wolfram-refine declined correctlynulldifference → -Cos[x] + Sqrt[1 - Sin[x]^2] · result → -Cos[x] + Sqrt[1 - Sin[x]^2]0.0222026-10-13 17:21:12 7a1dcfa636edElement[x, Reals] ⊢ (Sqrt[(1 - (Sin[x]*Sin[x]))] == Cos[x])null (correct: contingent)
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, Reals]);
  Residual = (Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x])→-Cos[x] + Sqrt[1 - Sin[x]^2]
resultRefine[Residual, Assum]→-Cos[x] + Sqrt[1 - Sin[x]^2]
answernull
wolfram-simplify declined correctlynulldifference → -Cos[x] + Sqrt[1 - Sin[x]^2] · result → Abs[Cos[x]] - Cos[x]0.0472026-10-13 17:21:08 7a1dcfa636edElement[x, Reals] ⊢ (Sqrt[(1 - (Sin[x]*Sin[x]))] == Cos[x])null (correct: contingent)
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, Reals]);
  Residual = (Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x])→-Cos[x] + Sqrt[1 - Sin[x]^2]
resultSimplify[Residual, Assum]→Abs[Cos[x]] - Cos[x]
answernull
Older runs (11)
systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
wolfram-fullsimplify no verdictnulldifference → -Cos[x] + Sqrt[1 - Sin[x]^2] · result → Abs[Cos[x]] - Cos[x]0.0972026-10-13 16:39:24 1e4dbd2f7d86Element[x, Reals] ⊢ (Sqrt[(1 - (Sin[x]*Sin[x]))] == 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, Reals]);
  Residual = (Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x])→-Cos[x] + Sqrt[1 - Sin[x]^2]
resultFullSimplify[Residual, Assum]→Abs[Cos[x]] - Cos[x]
answernull
wolfram-refine no verdictnulldifference → -Cos[x] + Sqrt[1 - Sin[x]^2] · result → -Cos[x] + Sqrt[1 - Sin[x]^2]0.0222026-10-13 16:39:21 1e4dbd2f7d86Element[x, Reals] ⊢ (Sqrt[(1 - (Sin[x]*Sin[x]))] == 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, Reals]);
  Residual = (Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x])→-Cos[x] + Sqrt[1 - Sin[x]^2]
resultRefine[Residual, Assum]→-Cos[x] + Sqrt[1 - Sin[x]^2]
answernull
wolfram-simplify no verdictnulldifference → -Cos[x] + Sqrt[1 - Sin[x]^2] · result → Abs[Cos[x]] - Cos[x]0.0482026-10-13 16:39:18 1e4dbd2f7d86Element[x, Reals] ⊢ (Sqrt[(1 - (Sin[x]*Sin[x]))] == 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, Reals]);
  Residual = (Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x])→-Cos[x] + Sqrt[1 - Sin[x]^2]
resultSimplify[Residual, Assum]→Abs[Cos[x]] - Cos[x]
answernull
sympy-simplifyno verdictnulldifference → sqrt(1 - sin(x)**2) - cos(x) · simplify → -cos(x) + Abs(cos(x))0.062026-10-13 16:39:17 1e4dbd2f7d86Q.real(x) ⊢ Eq(sqrt((1 - (sin(x)*sin(x)))), 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 = 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', real=True)
Residual = sqrt(1 - sin(x) * sin(x)) - 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')
differencesqrt(1 - sin(x) * sin(x)) - cos(x)→sqrt(1 - sin(x)**2) - cos(x)
simplifysimplify(Residual)→-cos(x) + Abs(cos(x))
answernull
sympy-refineno verdictnulldifference → sqrt(1 - sin(x)**2) - cos(x) · refine → sqrt(1 - sin(x)**2) - cos(x)0.0532026-10-13 16:39:16 1e4dbd2f7d86Q.real(x) ⊢ Eq(sqrt((1 - (sin(x)*sin(x)))), 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', real=True)
Assum = And(Q.real(x))
Residual = sqrt(1 - sin(x) * sin(x)) - 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')
differencesqrt(1 - sin(x) * sin(x)) - cos(x)→sqrt(1 - sin(x)**2) - cos(x)
refinerefine(Residual, Assum)→sqrt(1 - sin(x)**2) - cos(x)
answernull
always-noneno verdictnull0.02026-10-13 16:39:13 1e4dbd2f7d86(real x) ⊢ (= (sqrt (- 1 (* (sin x) (sin x)))) (cos x))null
sympy-askno verdictnulldifference → sqrt(1 - sin(x)**2) - cos(x) · ask → None0.412026-10-13 16:39:13 1e4dbd2f7d86Q.real(x) ⊢ Eq(sqrt((1 - (sin(x)*sin(x)))), 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', real=True)
Assum = And(Q.real(x))
Residual = sqrt(1 - sin(x) * sin(x)) - 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')
differencesqrt(1 - sin(x) * sin(x)) - cos(x)→sqrt(1 - sin(x)**2) - cos(x)
askQ.zero(Residual)→None
answernull
wolfram-fullsimplify no verdictnulldifference → -Cos[x] + Sqrt[1 - Sin[x]^2] · result → Abs[Cos[x]] - Cos[x]0.0952026-10-12 20:21:54 45d6b9056f06Element[x, Reals] ⊢ (Sqrt[(1 - (Sin[x]*Sin[x]))] == 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, Reals]);
  Residual = (Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x])→-Cos[x] + Sqrt[1 - Sin[x]^2]
resultFullSimplify[Residual, Assum]→Abs[Cos[x]] - Cos[x]
answernull
wolfram-refine no verdictnulldifference → -Cos[x] + Sqrt[1 - Sin[x]^2] · result → -Cos[x] + Sqrt[1 - Sin[x]^2]0.0222026-10-12 20:21:51 45d6b9056f06Element[x, Reals] ⊢ (Sqrt[(1 - (Sin[x]*Sin[x]))] == 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, Reals]);
  Residual = (Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x])→-Cos[x] + Sqrt[1 - Sin[x]^2]
resultRefine[Residual, Assum]→-Cos[x] + Sqrt[1 - Sin[x]^2]
answernull
always-noneno verdictnull0.02026-10-12 20:21:48 45d6b9056f06(real x) ⊢ (= (sqrt (- 1 (* (sin x) (sin x)))) (cos x))null
wolfram-simplify no verdictnulldifference → -Cos[x] + Sqrt[1 - Sin[x]^2] · result → Abs[Cos[x]] - Cos[x]0.0472026-10-12 20:21:48 45d6b9056f06Element[x, Reals] ⊢ (Sqrt[(1 - (Sin[x]*Sin[x]))] == 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, Reals]);
  Residual = (Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sqrt[1 - Sin[x]*Sin[x]]) - (Cos[x])→-Cos[x] + Sqrt[1 - Sin[x]^2]
resultSimplify[Residual, Assum]→Abs[Cos[x]] - Cos[x]
answernull

Source (trig.sexp)

(case trig-sqrt-01 (category identities)
  (note "sqrt(cos^2) is |cos x|, which is cos x only where cos x >= 0.")
  (var x real)
  (ask (= (sqrt (- 1 (* (sin x) (sin x)))) (cos x)))
  (expect contingent)
  (witness true (x 0))
  (witness false (x 3)))