← suite trig

trig-pyth-01 true

category identities

Summary

Given
\(z \in \mathbb{C}\)
Claim
\(\displaystyle \sin z \, \sin z + \cos z \, \cos z = 1\)
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)

systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
none
always-noneno verdictnull0.02026-10-13 17:21:03 7a1dcfa636ed(= (+ (* (sin z) (sin z)) (* (cos z) (cos z))) 1)null
sympy
sympy-askno verdictnulldifference → sin(z)**2 + cos(z)**2 - 1 · ask → None0.0862026-10-13 17:21:03 7a1dcfa636edQ.complex(z) ⊢ Eq(((sin(z)*sin(z)) + (cos(z)*cos(z))), 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()

z = Symbol('z', complex=True)
Assum = And(Q.complex(z))
Residual = sin(z) * sin(z) + cos(z) * cos(z) - 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')
differencesin(z) * sin(z) + cos(z) * cos(z) - 1→sin(z)**2 + cos(z)**2 - 1
askQ.zero(Residual)→None
answernull
sympy-askno verdictnulldifference → sin(z)**2 + cos(z)**2 - 1 · ask → None0.0292026-10-12 20:21:44 45d6b9056f06Q.complex(z) ⊢ Eq(((sin(z)*sin(z)) + (cos(z)*cos(z))), 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()

z = Symbol('z', complex=True)
Assum = And(Q.complex(z))
Residual = sin(z) * sin(z) + cos(z) * cos(z) - 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')
differencesin(z) * sin(z) + cos(z) * cos(z) - 1→sin(z)**2 + cos(z)**2 - 1
askQ.zero(Residual)→None
answernull
sympy-refineno verdictnulldifference → sin(z)**2 + cos(z)**2 - 1 · refine → sin(z)**2 + cos(z)**2 - 10.0482026-10-13 17:21:06 7a1dcfa636edQ.complex(z) ⊢ Eq(((sin(z)*sin(z)) + (cos(z)*cos(z))), 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()

z = Symbol('z', complex=True)
Assum = And(Q.complex(z))
Residual = sin(z) * sin(z) + cos(z) * cos(z) - 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')
differencesin(z) * sin(z) + cos(z) * cos(z) - 1→sin(z)**2 + cos(z)**2 - 1
refinerefine(Residual, Assum)→sin(z)**2 + cos(z)**2 - 1
answernull
sympy-refineno verdictnulldifference → sin(z)**2 + cos(z)**2 - 1 · refine → sin(z)**2 + cos(z)**2 - 10.052026-10-12 20:21:45 45d6b9056f06Q.complex(z) ⊢ Eq(((sin(z)*sin(z)) + (cos(z)*cos(z))), 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()

z = Symbol('z', complex=True)
Assum = And(Q.complex(z))
Residual = sin(z) * sin(z) + cos(z) * cos(z) - 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')
differencesin(z) * sin(z) + cos(z) * cos(z) - 1→sin(z)**2 + cos(z)**2 - 1
refinerefine(Residual, Assum)→sin(z)**2 + cos(z)**2 - 1
answernull
sympy-simplifysolvedtruedifference → sin(z)**2 + cos(z)**2 - 1 · simplify → 00.0352026-10-13 17:21:07 7a1dcfa636edQ.complex(z) ⊢ Eq(((sin(z)*sin(z)) + (cos(z)*cos(z))), 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()

z = Symbol('z', complex=True)
Residual = sin(z) * sin(z) + cos(z) * cos(z) - 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')
differencesin(z) * sin(z) + cos(z) * cos(z) - 1→sin(z)**2 + cos(z)**2 - 1
simplifysimplify(Residual)→0
answertrue
sympy-simplifysolvedtruedifference → sin(z)**2 + cos(z)**2 - 1 · simplify → 00.0362026-10-12 20:21:47 45d6b9056f06Q.complex(z) ⊢ Eq(((sin(z)*sin(z)) + (cos(z)*cos(z))), 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()

z = Symbol('z', complex=True)
Residual = sin(z) * sin(z) + cos(z) * cos(z) - 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')
differencesin(z) * sin(z) + cos(z) * cos(z) - 1→sin(z)**2 + cos(z)**2 - 1
simplifysimplify(Residual)→0
answertrue
wolfram
wolfram-fullsimplify solvedtruedifference → -1 + Cos[z]^2 + Sin[z]^2 · result → 00.0952026-10-13 17:21:15 7a1dcfa636edElement[z, Complexes] ⊢ (((Sin[z]*Sin[z]) + (Cos[z]*Cos[z])) == 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[z, Complexes]);
  Residual = (Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1)→-1 + Cos[z]^2 + Sin[z]^2
resultFullSimplify[Residual, Assum]→0
answertrue
wolfram-refine no verdictnulldifference → -1 + Cos[z]^2 + Sin[z]^2 · result → -1 + Cos[z]^2 + Sin[z]^20.0952026-10-13 17:21:12 7a1dcfa636edElement[z, Complexes] ⊢ (((Sin[z]*Sin[z]) + (Cos[z]*Cos[z])) == 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[z, Complexes]);
  Residual = (Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1)→-1 + Cos[z]^2 + Sin[z]^2
resultRefine[Residual, Assum]→-1 + Cos[z]^2 + Sin[z]^2
answernull
wolfram-simplify solvedtruedifference → -1 + Cos[z]^2 + Sin[z]^2 · result → 00.0952026-10-13 17:21:08 7a1dcfa636edElement[z, Complexes] ⊢ (((Sin[z]*Sin[z]) + (Cos[z]*Cos[z])) == 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[z, Complexes]);
  Residual = (Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1)→-1 + Cos[z]^2 + Sin[z]^2
resultSimplify[Residual, Assum]→0
answertrue
Older runs (11)
systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
wolfram-fullsimplify solvedtruedifference → -1 + Cos[z]^2 + Sin[z]^2 · result → 00.0962026-10-13 16:39:24 1e4dbd2f7d86Element[z, Complexes] ⊢ (((Sin[z]*Sin[z]) + (Cos[z]*Cos[z])) == 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[z, Complexes]);
  Residual = (Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1)→-1 + Cos[z]^2 + Sin[z]^2
resultFullSimplify[Residual, Assum]→0
answertrue
wolfram-refine no verdictnulldifference → -1 + Cos[z]^2 + Sin[z]^2 · result → -1 + Cos[z]^2 + Sin[z]^20.0952026-10-13 16:39:21 1e4dbd2f7d86Element[z, Complexes] ⊢ (((Sin[z]*Sin[z]) + (Cos[z]*Cos[z])) == 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[z, Complexes]);
  Residual = (Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1)→-1 + Cos[z]^2 + Sin[z]^2
resultRefine[Residual, Assum]→-1 + Cos[z]^2 + Sin[z]^2
answernull
wolfram-simplify solvedtruedifference → -1 + Cos[z]^2 + Sin[z]^2 · result → 00.0962026-10-13 16:39:18 1e4dbd2f7d86Element[z, Complexes] ⊢ (((Sin[z]*Sin[z]) + (Cos[z]*Cos[z])) == 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[z, Complexes]);
  Residual = (Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1)→-1 + Cos[z]^2 + Sin[z]^2
resultSimplify[Residual, Assum]→0
answertrue
sympy-simplifysolvedtruedifference → sin(z)**2 + cos(z)**2 - 1 · simplify → 00.0352026-10-13 16:39:17 1e4dbd2f7d86Q.complex(z) ⊢ Eq(((sin(z)*sin(z)) + (cos(z)*cos(z))), 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()

z = Symbol('z', complex=True)
Residual = sin(z) * sin(z) + cos(z) * cos(z) - 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')
differencesin(z) * sin(z) + cos(z) * cos(z) - 1→sin(z)**2 + cos(z)**2 - 1
simplifysimplify(Residual)→0
answertrue
sympy-refineno verdictnulldifference → sin(z)**2 + cos(z)**2 - 1 · refine → sin(z)**2 + cos(z)**2 - 10.0482026-10-13 16:39:16 1e4dbd2f7d86Q.complex(z) ⊢ Eq(((sin(z)*sin(z)) + (cos(z)*cos(z))), 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()

z = Symbol('z', complex=True)
Assum = And(Q.complex(z))
Residual = sin(z) * sin(z) + cos(z) * cos(z) - 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')
differencesin(z) * sin(z) + cos(z) * cos(z) - 1→sin(z)**2 + cos(z)**2 - 1
refinerefine(Residual, Assum)→sin(z)**2 + cos(z)**2 - 1
answernull
always-noneno verdictnull0.02026-10-13 16:39:13 1e4dbd2f7d86(= (+ (* (sin z) (sin z)) (* (cos z) (cos z))) 1)null
sympy-askno verdictnulldifference → sin(z)**2 + cos(z)**2 - 1 · ask → None0.0882026-10-13 16:39:13 1e4dbd2f7d86Q.complex(z) ⊢ Eq(((sin(z)*sin(z)) + (cos(z)*cos(z))), 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()

z = Symbol('z', complex=True)
Assum = And(Q.complex(z))
Residual = sin(z) * sin(z) + cos(z) * cos(z) - 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')
differencesin(z) * sin(z) + cos(z) * cos(z) - 1→sin(z)**2 + cos(z)**2 - 1
askQ.zero(Residual)→None
answernull
wolfram-fullsimplify solvedtruedifference → -1 + Cos[z]^2 + Sin[z]^2 · result → 00.0952026-10-12 20:21:54 45d6b9056f06Element[z, Complexes] ⊢ (((Sin[z]*Sin[z]) + (Cos[z]*Cos[z])) == 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[z, Complexes]);
  Residual = (Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1)→-1 + Cos[z]^2 + Sin[z]^2
resultFullSimplify[Residual, Assum]→0
answertrue
wolfram-refine no verdictnulldifference → -1 + Cos[z]^2 + Sin[z]^2 · result → -1 + Cos[z]^2 + Sin[z]^20.0952026-10-12 20:21:51 45d6b9056f06Element[z, Complexes] ⊢ (((Sin[z]*Sin[z]) + (Cos[z]*Cos[z])) == 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[z, Complexes]);
  Residual = (Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1)→-1 + Cos[z]^2 + Sin[z]^2
resultRefine[Residual, Assum]→-1 + Cos[z]^2 + Sin[z]^2
answernull
always-noneno verdictnull0.02026-10-12 20:21:48 45d6b9056f06(= (+ (* (sin z) (sin z)) (* (cos z) (cos z))) 1)null
wolfram-simplify solvedtruedifference → -1 + Cos[z]^2 + Sin[z]^2 · result → 00.0952026-10-12 20:21:48 45d6b9056f06Element[z, Complexes] ⊢ (((Sin[z]*Sin[z]) + (Cos[z]*Cos[z])) == 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[z, Complexes]);
  Residual = (Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Sin[z]*Sin[z] + Cos[z]*Cos[z]) - (1)→-1 + Cos[z]^2 + Sin[z]^2
resultSimplify[Residual, Assum]→0
answertrue

Source (trig.sexp)

(case trig-pyth-01 (category identities)
  (var z complex)
  (ask (= (+ (* (sin z) (sin z)) (* (cos z) (cos z))) 1))
  (expect true))