demo-eq-trig-01 true

category trig

Summary

Given

\(x \in \mathbb{C}\)

Claim

\(\displaystyle \left(\sin x\right)^{2} + \left(\cos x\right)^{2} = 1\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

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 17:20:05 df65d778fc5c (= (+ (pow (sin x) 2) (pow (cos x) 2)) 1) null

sympy

sympy-ask no verdict null difference → sin(x)**2 + cos(x)**2 - 1 · ask → None 0.086 2026-10-13 17:20:46 df65d778fc5c Q.complex(x) ⊢ Eq((Pow(sin(x), 2) + Pow(cos(x), 2)), 1) 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))
Residual = Pow(sin(x), 2) + Pow(cos(x), 2) - 1
# 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	`Pow(sin(x), 2) + Pow(cos(x), 2) - 1`	→	`sin(x)2 + cos(x)2 - 1`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → sin(x)**2 + cos(x)**2 - 1 · ask → None 0.028 2026-10-12 20:21:28 d7af4def9aaa Q.complex(x) ⊢ Eq((Pow(sin(x), 2) + Pow(cos(x), 2)), 1) 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))
Residual = Pow(sin(x), 2) + Pow(cos(x), 2) - 1
# 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	`Pow(sin(x), 2) + Pow(cos(x), 2) - 1`	→	`sin(x)2 + cos(x)2 - 1`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine no verdict null difference → sin(x)**2 + cos(x)**2 - 1 · refine → sin(x)**2 + cos(x)**2 - 1 0.047 2026-10-13 17:20:49 df65d778fc5c Q.complex(x) ⊢ Eq((Pow(sin(x), 2) + Pow(cos(x), 2)), 1) 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))
Residual = Pow(sin(x), 2) + Pow(cos(x), 2) - 1
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	`Pow(sin(x), 2) + Pow(cos(x), 2) - 1`	→	`sin(x)2 + cos(x)2 - 1`
refine	`refine(Residual, Assum)`	→	`sin(x)2 + cos(x)2 - 1`
answer			`null`

sympy-refine no verdict null difference → sin(x)**2 + cos(x)**2 - 1 · refine → sin(x)**2 + cos(x)**2 - 1 0.049 2026-10-12 20:21:30 d7af4def9aaa Q.complex(x) ⊢ Eq((Pow(sin(x), 2) + Pow(cos(x), 2)), 1) 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))
Residual = Pow(sin(x), 2) + Pow(cos(x), 2) - 1
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	`Pow(sin(x), 2) + Pow(cos(x), 2) - 1`	→	`sin(x)2 + cos(x)2 - 1`
refine	`refine(Residual, Assum)`	→	`sin(x)2 + cos(x)2 - 1`
answer			`null`

sympy-simplify solved true difference → sin(x)**2 + cos(x)**2 - 1 · simplify → 0 0.034 2026-10-13 17:20:51 df65d778fc5c Q.complex(x) ⊢ Eq((Pow(sin(x), 2) + Pow(cos(x), 2)), 1) 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)
Residual = Pow(sin(x), 2) + Pow(cos(x), 2) - 1
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	`Pow(sin(x), 2) + Pow(cos(x), 2) - 1`	→	`sin(x)2 + cos(x)2 - 1`
simplify	`simplify(Residual)`	→	`0`
answer			`true`

sympy-simplify solved true difference → sin(x)**2 + cos(x)**2 - 1 · simplify → 0 0.036 2026-10-12 20:21:32 d7af4def9aaa Q.complex(x) ⊢ Eq((Pow(sin(x), 2) + Pow(cos(x), 2)), 1) 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)
Residual = Pow(sin(x), 2) + Pow(cos(x), 2) - 1
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	`Pow(sin(x), 2) + Pow(cos(x), 2) - 1`	→	`sin(x)2 + cos(x)2 - 1`
simplify	`simplify(Residual)`	→	`0`
answer			`true`

wolfram

wolfram-fullsimplify solved true difference → -1 + Cos[x]^2 + Sin[x]^2 · result → 0 0.021 2026-10-13 17:20:59 df65d778fc5c Element[x, Complexes] ⊢ ((Power[Sin[x], 2] + Power[Cos[x], 2]) == 1) true

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]);
  Residual = (Sin[x]^2 + Cos[x]^2) - (1);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[x]^2 + Cos[x]^2) - (1)`	→	`-1 + Cos[x]^2 + Sin[x]^2`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine no verdict null difference → -1 + Cos[x]^2 + Sin[x]^2 · result → -1 + Cos[x]^2 + Sin[x]^2 0.021 2026-10-13 17:20:56 df65d778fc5c Element[x, Complexes] ⊢ ((Power[Sin[x], 2] + Power[Cos[x], 2]) == 1) 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]);
  Residual = (Sin[x]^2 + Cos[x]^2) - (1);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[x]^2 + Cos[x]^2) - (1)`	→	`-1 + Cos[x]^2 + Sin[x]^2`
result	`Refine[Residual, Assum]`	→	`-1 + Cos[x]^2 + Sin[x]^2`
answer			`null`

wolfram-simplify solved true difference → -1 + Cos[x]^2 + Sin[x]^2 · result → 0 0.021 2026-10-13 17:20:52 df65d778fc5c Element[x, Complexes] ⊢ ((Power[Sin[x], 2] + Power[Cos[x], 2]) == 1) true

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]);
  Residual = (Sin[x]^2 + Cos[x]^2) - (1);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[x]^2 + Cos[x]^2) - (1)`	→	`-1 + Cos[x]^2 + Sin[x]^2`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

Older runs (11)

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

wolfram-fullsimplify solved true difference → -1 + Cos[x]^2 + Sin[x]^2 · result → 0 0.021 2026-10-13 16:39:09 03f5149718d0 Element[x, Complexes] ⊢ ((Power[Sin[x], 2] + Power[Cos[x], 2]) == 1) true

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]);
  Residual = (Sin[x]^2 + Cos[x]^2) - (1);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[x]^2 + Cos[x]^2) - (1)`	→	`-1 + Cos[x]^2 + Sin[x]^2`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine no verdict null difference → -1 + Cos[x]^2 + Sin[x]^2 · result → -1 + Cos[x]^2 + Sin[x]^2 0.021 2026-10-13 16:39:06 03f5149718d0 Element[x, Complexes] ⊢ ((Power[Sin[x], 2] + Power[Cos[x], 2]) == 1) 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]);
  Residual = (Sin[x]^2 + Cos[x]^2) - (1);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[x]^2 + Cos[x]^2) - (1)`	→	`-1 + Cos[x]^2 + Sin[x]^2`
result	`Refine[Residual, Assum]`	→	`-1 + Cos[x]^2 + Sin[x]^2`
answer			`null`

wolfram-simplify solved true difference → -1 + Cos[x]^2 + Sin[x]^2 · result → 0 0.022 2026-10-13 16:39:02 03f5149718d0 Element[x, Complexes] ⊢ ((Power[Sin[x], 2] + Power[Cos[x], 2]) == 1) true

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]);
  Residual = (Sin[x]^2 + Cos[x]^2) - (1);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[x]^2 + Cos[x]^2) - (1)`	→	`-1 + Cos[x]^2 + Sin[x]^2`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

sympy-simplify solved true difference → sin(x)**2 + cos(x)**2 - 1 · simplify → 0 0.035 2026-10-13 16:39:01 03f5149718d0 Q.complex(x) ⊢ Eq((Pow(sin(x), 2) + Pow(cos(x), 2)), 1) 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)
Residual = Pow(sin(x), 2) + Pow(cos(x), 2) - 1
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	`Pow(sin(x), 2) + Pow(cos(x), 2) - 1`	→	`sin(x)2 + cos(x)2 - 1`
simplify	`simplify(Residual)`	→	`0`
answer			`true`

sympy-refine no verdict null difference → sin(x)**2 + cos(x)**2 - 1 · refine → sin(x)**2 + cos(x)**2 - 1 0.047 2026-10-13 16:38:59 03f5149718d0 Q.complex(x) ⊢ Eq((Pow(sin(x), 2) + Pow(cos(x), 2)), 1) 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))
Residual = Pow(sin(x), 2) + Pow(cos(x), 2) - 1
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	`Pow(sin(x), 2) + Pow(cos(x), 2) - 1`	→	`sin(x)2 + cos(x)2 - 1`
refine	`refine(Residual, Assum)`	→	`sin(x)2 + cos(x)2 - 1`
answer			`null`

always-none no verdict null 0.0 2026-10-13 16:38:56 03f5149718d0 (= (+ (pow (sin x) 2) (pow (cos x) 2)) 1) null

sympy-ask no verdict null difference → sin(x)**2 + cos(x)**2 - 1 · ask → None 0.086 2026-10-13 16:38:56 03f5149718d0 Q.complex(x) ⊢ Eq((Pow(sin(x), 2) + Pow(cos(x), 2)), 1) 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))
Residual = Pow(sin(x), 2) + Pow(cos(x), 2) - 1
# 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	`Pow(sin(x), 2) + Pow(cos(x), 2) - 1`	→	`sin(x)2 + cos(x)2 - 1`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify solved true difference → -1 + Cos[x]^2 + Sin[x]^2 · result → 0 0.021 2026-10-12 20:21:41 d7af4def9aaa Element[x, Complexes] ⊢ ((Power[Sin[x], 2] + Power[Cos[x], 2]) == 1) true

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]);
  Residual = (Sin[x]^2 + Cos[x]^2) - (1);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[x]^2 + Cos[x]^2) - (1)`	→	`-1 + Cos[x]^2 + Sin[x]^2`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine no verdict null difference → -1 + Cos[x]^2 + Sin[x]^2 · result → -1 + Cos[x]^2 + Sin[x]^2 0.021 2026-10-12 20:21:37 d7af4def9aaa Element[x, Complexes] ⊢ ((Power[Sin[x], 2] + Power[Cos[x], 2]) == 1) 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]);
  Residual = (Sin[x]^2 + Cos[x]^2) - (1);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[x]^2 + Cos[x]^2) - (1)`	→	`-1 + Cos[x]^2 + Sin[x]^2`
result	`Refine[Residual, Assum]`	→	`-1 + Cos[x]^2 + Sin[x]^2`
answer			`null`

wolfram-simplify solved true difference → -1 + Cos[x]^2 + Sin[x]^2 · result → 0 0.021 2026-10-12 20:21:34 d7af4def9aaa Element[x, Complexes] ⊢ ((Power[Sin[x], 2] + Power[Cos[x], 2]) == 1) true

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]);
  Residual = (Sin[x]^2 + Cos[x]^2) - (1);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[x]^2 + Cos[x]^2) - (1)`	→	`-1 + Cos[x]^2 + Sin[x]^2`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

always-none no verdict null 0.0 2026-10-12 20:21:33 d7af4def9aaa (= (+ (pow (sin x) 2) (pow (cos x) 2)) 1) null

Source (equalities.sexp)

(case demo-eq-trig-01 (category trig)
  (var x complex)
  (ask (= (+ (pow (sin x) 2) (pow (cos x) 2)) 1))
  (expect true))