trig-double-02 false

category identities

Summary

Given

\(x \in \mathbb{R},\quad x > 0,\quad x < 1\)

Claim

\(\displaystyle \sin\left(2 \, x\right) \neq 2 \, \sin x\)

Expected

false unequal at every admissible point (e.g. at \(x = \frac{1}{2}\))

the two sides differ at every admissible point

witness (false): \(x = \frac{1}{2}\)

sin 2x = 2 sin x iff sin x (cos x - 1) = 0: never on (0, 1).

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 17:21:03 7a1dcfa636ed (real x) ∧ (positive x) ∧ (< x 1) ⊢ (= (sin (* 2 x)) (* 2 (sin x))) null

sympy

sympy-ask no verdict null difference → -2*sin(x) + sin(2*x) · ask → None 0.088 2026-10-13 17:21:03 7a1dcfa636ed Q.real(x) ∧ Q.positive(x) ∧ (x < 1) ⊢ Eq(sin((2*x)), (2*sin(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), Q.positive(x), x < 1)
Residual = sin(2 * x) - (2 * sin(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	`sin(2 * x) - (2 * sin(x))`	→	`-2sin(x) + sin(2x)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → -2*sin(x) + sin(2*x) · ask → None 0.031 2026-10-12 20:21:44 45d6b9056f06 Q.real(x) ∧ Q.positive(x) ∧ (x < 1) ⊢ Eq(sin((2*x)), (2*sin(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), Q.positive(x), x < 1)
Residual = sin(2 * x) - (2 * sin(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	`sin(2 * x) - (2 * sin(x))`	→	`-2sin(x) + sin(2x)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine no verdict null difference → -2*sin(x) + sin(2*x) · refine → -2*sin(x) + sin(2*x) 0.038 2026-10-13 17:21:06 7a1dcfa636ed Q.real(x) ∧ Q.positive(x) ∧ (x < 1) ⊢ Eq(sin((2*x)), (2*sin(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), Q.positive(x), x < 1)
Residual = sin(2 * x) - (2 * sin(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	`sin(2 * x) - (2 * sin(x))`	→	`-2sin(x) + sin(2x)`
refine	`refine(Residual, Assum)`	→	`-2sin(x) + sin(2x)`
answer			`null`

sympy-refine no verdict null difference → -2*sin(x) + sin(2*x) · refine → -2*sin(x) + sin(2*x) 0.044 2026-10-12 20:21:45 45d6b9056f06 Q.real(x) ∧ Q.positive(x) ∧ (x < 1) ⊢ Eq(sin((2*x)), (2*sin(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), Q.positive(x), x < 1)
Residual = sin(2 * x) - (2 * sin(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	`sin(2 * x) - (2 * sin(x))`	→	`-2sin(x) + sin(2x)`
refine	`refine(Residual, Assum)`	→	`-2sin(x) + sin(2x)`
answer			`null`

sympy-simplify no verdict null difference → -2*sin(x) + sin(2*x) · simplify → 2*(cos(x) - 1)*sin(x) 0.041 2026-10-13 17:21:07 7a1dcfa636ed Q.real(x) ∧ Q.positive(x) ∧ (x < 1) ⊢ Eq(sin((2*x)), (2*sin(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, positive=True)
# dropped (not a unary fact about one symbol): x < 1
Residual = sin(2 * x) - (2 * sin(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	`sin(2 * x) - (2 * sin(x))`	→	`-2sin(x) + sin(2x)`
simplify	`simplify(Residual)`	→	`2(cos(x) - 1)sin(x)`
answer			`null`

sympy-simplify no verdict null difference → -2*sin(x) + sin(2*x) · simplify → 2*(cos(x) - 1)*sin(x) 0.045 2026-10-12 20:21:47 45d6b9056f06 Q.real(x) ∧ Q.positive(x) ∧ (x < 1) ⊢ Eq(sin((2*x)), (2*sin(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, positive=True)
# dropped (not a unary fact about one symbol): x < 1
Residual = sin(2 * x) - (2 * sin(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	`sin(2 * x) - (2 * sin(x))`	→	`-2sin(x) + sin(2x)`
simplify	`simplify(Residual)`	→	`2(cos(x) - 1)sin(x)`
answer			`null`

wolfram

wolfram-fullsimplify no verdict null difference → -2*Sin[x] + Sin[2*x] · result → -2*Sin[x] + Sin[2*x] 0.043 2026-10-13 17:21:15 7a1dcfa636ed Element[x, Reals] ∧ (Element[x, Reals] && x > 0) ∧ (x < 1) ⊢ (Sin[(2*x)] == (2*Sin[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]) && ((Element[x, Reals] && x > 0)) && ((x < 1));
  Residual = (Sin[2*x]) - (2*Sin[x]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[2x]) - (2Sin[x])`	→	`-2Sin[x] + Sin[2x]`
result	`FullSimplify[Residual, Assum]`	→	`-2Sin[x] + Sin[2x]`
answer			`null`

wolfram-refine no verdict null difference → -2*Sin[x] + Sin[2*x] · result → -2*Sin[x] + Sin[2*x] 0.021 2026-10-13 17:21:12 7a1dcfa636ed Element[x, Reals] ∧ (Element[x, Reals] && x > 0) ∧ (x < 1) ⊢ (Sin[(2*x)] == (2*Sin[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]) && ((Element[x, Reals] && x > 0)) && ((x < 1));
  Residual = (Sin[2*x]) - (2*Sin[x]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

wolfram-simplify no verdict null difference → -2*Sin[x] + Sin[2*x] · result → -2*Sin[x] + Sin[2*x] 0.024 2026-10-13 17:21:08 7a1dcfa636ed Element[x, Reals] ∧ (Element[x, Reals] && x > 0) ∧ (x < 1) ⊢ (Sin[(2*x)] == (2*Sin[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]) && ((Element[x, Reals] && x > 0)) && ((x < 1));
  Residual = (Sin[2*x]) - (2*Sin[x]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[2x]) - (2Sin[x])`	→	`-2Sin[x] + Sin[2x]`
result	`Simplify[Residual, Assum]`	→	`-2Sin[x] + Sin[2x]`
answer			`null`

Older runs (11)

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

wolfram-fullsimplify no verdict null difference → -2*Sin[x] + Sin[2*x] · result → -2*Sin[x] + Sin[2*x] 0.091 2026-10-13 16:39:24 1e4dbd2f7d86 Element[x, Reals] ∧ (Element[x, Reals] && x > 0) ∧ (x < 1) ⊢ (Sin[(2*x)] == (2*Sin[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]) && ((Element[x, Reals] && x > 0)) && ((x < 1));
  Residual = (Sin[2*x]) - (2*Sin[x]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[2x]) - (2Sin[x])`	→	`-2Sin[x] + Sin[2x]`
result	`FullSimplify[Residual, Assum]`	→	`-2Sin[x] + Sin[2x]`
answer			`null`

wolfram-refine no verdict null difference → -2*Sin[x] + Sin[2*x] · result → -2*Sin[x] + Sin[2*x] 0.022 2026-10-13 16:39:21 1e4dbd2f7d86 Element[x, Reals] ∧ (Element[x, Reals] && x > 0) ∧ (x < 1) ⊢ (Sin[(2*x)] == (2*Sin[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]) && ((Element[x, Reals] && x > 0)) && ((x < 1));
  Residual = (Sin[2*x]) - (2*Sin[x]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

wolfram-simplify no verdict null difference → -2*Sin[x] + Sin[2*x] · result → -2*Sin[x] + Sin[2*x] 0.024 2026-10-13 16:39:18 1e4dbd2f7d86 Element[x, Reals] ∧ (Element[x, Reals] && x > 0) ∧ (x < 1) ⊢ (Sin[(2*x)] == (2*Sin[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]) && ((Element[x, Reals] && x > 0)) && ((x < 1));
  Residual = (Sin[2*x]) - (2*Sin[x]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[2x]) - (2Sin[x])`	→	`-2Sin[x] + Sin[2x]`
result	`Simplify[Residual, Assum]`	→	`-2Sin[x] + Sin[2x]`
answer			`null`

sympy-simplify no verdict null difference → -2*sin(x) + sin(2*x) · simplify → 2*(cos(x) - 1)*sin(x) 0.042 2026-10-13 16:39:17 1e4dbd2f7d86 Q.real(x) ∧ Q.positive(x) ∧ (x < 1) ⊢ Eq(sin((2*x)), (2*sin(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, positive=True)
# dropped (not a unary fact about one symbol): x < 1
Residual = sin(2 * x) - (2 * sin(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	`sin(2 * x) - (2 * sin(x))`	→	`-2sin(x) + sin(2x)`
simplify	`simplify(Residual)`	→	`2(cos(x) - 1)sin(x)`
answer			`null`

sympy-refine no verdict null difference → -2*sin(x) + sin(2*x) · refine → -2*sin(x) + sin(2*x) 0.038 2026-10-13 16:39:16 1e4dbd2f7d86 Q.real(x) ∧ Q.positive(x) ∧ (x < 1) ⊢ Eq(sin((2*x)), (2*sin(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), Q.positive(x), x < 1)
Residual = sin(2 * x) - (2 * sin(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	`sin(2 * x) - (2 * sin(x))`	→	`-2sin(x) + sin(2x)`
refine	`refine(Residual, Assum)`	→	`-2sin(x) + sin(2x)`
answer			`null`

always-none no verdict null 0.0 2026-10-13 16:39:13 1e4dbd2f7d86 (real x) ∧ (positive x) ∧ (< x 1) ⊢ (= (sin (* 2 x)) (* 2 (sin x))) null

sympy-ask no verdict null difference → -2*sin(x) + sin(2*x) · ask → None 0.089 2026-10-13 16:39:13 1e4dbd2f7d86 Q.real(x) ∧ Q.positive(x) ∧ (x < 1) ⊢ Eq(sin((2*x)), (2*sin(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), Q.positive(x), x < 1)
Residual = sin(2 * x) - (2 * sin(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	`sin(2 * x) - (2 * sin(x))`	→	`-2sin(x) + sin(2x)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify no verdict null difference → -2*Sin[x] + Sin[2*x] · result → -2*Sin[x] + Sin[2*x] 0.045 2026-10-12 20:21:54 45d6b9056f06 Element[x, Reals] ∧ (Element[x, Reals] && x > 0) ∧ (x < 1) ⊢ (Sin[(2*x)] == (2*Sin[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]) && ((Element[x, Reals] && x > 0)) && ((x < 1));
  Residual = (Sin[2*x]) - (2*Sin[x]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[2x]) - (2Sin[x])`	→	`-2Sin[x] + Sin[2x]`
result	`FullSimplify[Residual, Assum]`	→	`-2Sin[x] + Sin[2x]`
answer			`null`

wolfram-refine no verdict null difference → -2*Sin[x] + Sin[2*x] · result → -2*Sin[x] + Sin[2*x] 0.021 2026-10-12 20:21:51 45d6b9056f06 Element[x, Reals] ∧ (Element[x, Reals] && x > 0) ∧ (x < 1) ⊢ (Sin[(2*x)] == (2*Sin[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]) && ((Element[x, Reals] && x > 0)) && ((x < 1));
  Residual = (Sin[2*x]) - (2*Sin[x]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

always-none no verdict null 0.0 2026-10-12 20:21:48 45d6b9056f06 (real x) ∧ (positive x) ∧ (< x 1) ⊢ (= (sin (* 2 x)) (* 2 (sin x))) null

wolfram-simplify no verdict null difference → -2*Sin[x] + Sin[2*x] · result → -2*Sin[x] + Sin[2*x] 0.024 2026-10-12 20:21:48 45d6b9056f06 Element[x, Reals] ∧ (Element[x, Reals] && x > 0) ∧ (x < 1) ⊢ (Sin[(2*x)] == (2*Sin[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]) && ((Element[x, Reals] && x > 0)) && ((x < 1));
  Residual = (Sin[2*x]) - (2*Sin[x]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[2x]) - (2Sin[x])`	→	`-2Sin[x] + Sin[2x]`
result	`Simplify[Residual, Assum]`	→	`-2Sin[x] + Sin[2x]`
answer			`null`

Source (trig.sexp)

(case trig-double-02 (category identities)
  (note "sin 2x = 2 sin x iff sin x (cos x - 1) = 0: never on (0, 1).")
  (var x real)
  (assume (positive x)) (assume (< x 1))
  (ask (= (sin (* 2 x)) (* 2 (sin x))))
  (expect false)
  (witness false (x 1/2)))