Trig Identity Verifier
Enter the two sides of a candidate identity as expressions in x. The verifier evaluates both at a battery of test points scattered across (0, 2π); if they agree everywhere to within tolerance the identity is reported as verified, otherwise a counterexample is shown.
Refutation is cheap, proof is not
Proving a trigonometric identity means transforming one side into the other through known identities, and there is no algorithm for it — the choice of which identity to apply next is genuinely a matter of insight. Disproving one, by contrast, takes a single value where the two sides disagree.
This verifier does the second reliably and offers strong evidence for the first. It evaluates both sides at a spread of test points and reports either agreement everywhere tested, or the specific value where they part company.
The asymmetry is the whole design. One disagreeing point settles the question conclusively against a candidate identity, and the tool reports exactly where — which is far more useful than a bare verdict, because it shows what to examine.
Agreement across many points is not a proof, and the output says so. But it catches essentially every real mistake: a dropped factor, a wrong sign, a misremembered double-angle formula all fail at the first or second sample.
How to use this calculator
- Enter the left-hand side An expression in x, using standard notation. Powers of a function are written by squaring the whole call, as in sin(x)^2.
- Enter the right-hand side The expression it is claimed to equal. A constant is a perfectly valid side.
- Read the verdict Either verified across the tested points, or not an identity — in which case a specific counterexample follows.
- Treat verification as evidence, not proof It means no disagreement was found at the sample points, which is strong but not conclusive.
How the check is performed
Both sides are compiled as functions of x and then evaluated at a fixed list of a dozen values scattered across one period, from just above zero to just under a full turn. The points are chosen to avoid the obvious singularities where tangent and the reciprocal functions blow up, so an identity involving them still gets a fair test.
At each point the two values are compared on a relative scale rather than an absolute one: they must differ by less than a ten-millionth of the larger of them. That scaling matters because expressions taking large values would otherwise fail on rounding alone, while expressions near zero would pass anything.
The comparison stops at the first disagreement, and that point becomes the reported counterexample with both sides' values shown. A single failure is enough — an identity must hold everywhere, so one exception disproves it.
Points where either side is undefined or non-finite are skipped rather than counted as failures. If fewer than four usable points remain, the verifier declines to give a verdict rather than pronouncing on almost no evidence.
What each input means
- LHS Left-hand side — form field “Left-hand side (in x)”
- An expression in x. It is compiled once and evaluated at each test point.
- RHS Right-hand side — form field “Right-hand side (in x)”
- The expression claimed to equal it. Which side is which makes no difference to the result.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
The Pythagorean identity
Sine squared plus cosine squared against 1 — the identity everything else in trigonometry is built on.
Inputs Left-hand side (in x) = sin(x)^2 + cos(x)^2, Right-hand side (in x) = 1
- LHS sin(x)^2 + cos(x)^2
- RHS 1
- Tested at sample points 12 test values across (0, 2π) all agree to within tolerance.
Result Identity verified
Every test point agrees, and the verdict is that the identity holds. This one is genuinely provable in a line from the definition of sine and cosine on the unit circle, so the numerical agreement is confirming something already certain.
Note the constant on the right is a valid expression. An identity does not require both sides to look complicated, only that they agree everywhere.
A double-angle identity
Sine of twice the angle against twice the product of sine and cosine — a standard result that is easy to misremember.
Inputs Left-hand side (in x) = sin(2*x), Right-hand side (in x) = 2*sin(x)*cos(x)
- LHS sin(2*x)
- RHS 2*sin(x)*cos(x)
- Tested at sample points 12 test values across (0, 2π) all agree to within tolerance.
Result Identity verified
The two sides agree at every point. This is the identity most often corrupted by dropping the factor of two, and the check would catch that immediately — the wrong version fails at the very first sample.
The two sides look nothing alike, which is exactly when numerical verification earns its place. Transforming one into the other symbolically takes several steps.
A candidate that fails
Sine against cosine. They agree at some angles but are not the same function.
Inputs Left-hand side (in x) = sin(x), Right-hand side (in x) = cos(x)
- LHS sin(x)
- RHS cos(x)
- Counterexample At x = 0.17: LHS = 0.169182, RHS = 0.985585 — the two sides disagree.
Result Not an identity
The verdict is that this is not an identity, and a specific value is given where the two differ along with both of their values there. That counterexample is the disproof — no further argument is needed.
Sine and cosine do coincide at certain angles, which is why agreeing at one point proves nothing. An identity must hold at every point, and the sampling is designed to find the disagreements quickly.
Reading the result
A counterexample is conclusive
One point where the sides differ disproves the identity outright, and nothing further is needed. That is why the check stops at the first disagreement rather than continuing.
Verified means not disproved
The tool has found no disagreement among its test points. Two functions could in principle agree at all of them and differ elsewhere, so this is evidence rather than proof — though in practice a wrong candidate almost never survives.
Undefined points are skipped, not failed
An identity involving tangent is undefined at certain angles, and both sides being undefined there is not a disagreement. Those points are passed over, and a verdict is withheld if too few usable ones remain.
When you would use this
Checking a candidate before attempting a proof
Time spent proving a false identity is wasted entirely. A few seconds here establishes whether the statement is worth the effort.
Locating an error in a transformation
Comparing an intermediate line of your working against the original left-hand side isolates the step where the two diverged, rather than showing only that the final answer is wrong.
Assumptions and limitations
What this calculator assumes
- Both sides are expressions in x that can be evaluated numerically.
- Agreement is judged relative to the magnitude of the values rather than by a fixed absolute tolerance.
- Test points are fixed and spread across one period, avoiding the obvious singularities.
- At least four usable points are required before a verdict is given.
Where it stops being the right tool
- Not a proof: agreement at finitely many points cannot establish equality everywhere.
- No symbolic manipulation, so no derivation or sequence of steps is produced.
- The sample points are fixed and cannot be chosen, so an identity failing only in a very narrow region could be missed.
- One variable, and identities in two angles cannot be tested.
Common mistakes
Treating verification as a proof
Why it happens. The verdict reads as definite, and for every identity a student is likely to meet it is correct.
How to avoid it. Use it to decide whether to attempt the proof, not to replace it. An exam asking you to prove an identity wants the transformation, not evidence.
Writing a squared function ambiguously
Why it happens. The conventional notation places the exponent on the function name, which no parser accepts. Writing it directly usually produces a different expression rather than an error.
How to avoid it. Square the whole call, as in sin(x)^2. If a verified identity comes back refuted, check this before doubting the mathematics.
Concluding from a single agreeing value
Why it happens. Two different functions frequently coincide somewhere, and finding one such point feels like confirmation.
How to avoid it. An identity must hold everywhere. That is why the check uses a spread of points and stops only when one disagrees.
Frequently asked questions
Is a verified identity really a proof?
No. Agreement at a dozen points is strong empirical evidence but not a formal proof — two functions could agree at all of them and differ elsewhere. A genuine proof transforms one side into the other through known identities, which is a separate and much harder task.
Why use sampling instead of symbolic manipulation?
Because trigonometric simplification has no canonical form and no reliable algorithm — the right next step is a matter of insight. Sampling catches essentially every real mistake quickly and, when a candidate fails, hands you a concrete counterexample.
What sample points are used?
About a dozen values spread across one period, from just above zero to just under a full turn, chosen to avoid the obvious singularities. The tolerance is relative, so the test scales with the size of the expressions rather than failing on large values.
What happens at points where an expression is undefined?
They are skipped rather than counted against the identity. If too few usable points remain — fewer than four — the verifier declines to give a verdict instead of pronouncing on almost no evidence.