{
    "name": "Trigonometric Identity Checker",
    "url": "https://calculators.geekmidst.com/calculators/trigonometric-identity-verification-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/trigonometric-identity-verification-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Type both sides of a trigonometric identity and test them numerically at 400 angles to see whether they agree, with a counterexample if they do not.",
    "summary": "Enter the left and right sides as expressions in x. The checker evaluates both at hundreds of angles: a single mismatch proves the identity false, while agreement everywhere is strong evidence (not a proof) that it is true.",
    "also_called": [
        "Trig identity verifier"
    ],
    "applies_to": "Worldwide",
    "last_updated": "2026-10-29",
    "computation": "Results are computed in the visitor’s browser on the HTML page. This API describes the calculator; it does not compute results.",
    "inputs": [
        {
            "name": "lhs",
            "label": "Left side",
            "type": "text",
            "default": "cos(2x)"
        },
        {
            "name": "rhs",
            "label": "Right side",
            "type": "text",
            "default": "1 - 2sin^2(x)"
        },
        {
            "name": "unit",
            "label": "Angles in",
            "type": "segmented",
            "default": "deg",
            "options": {
                "deg": "Degrees",
                "rad": "Radians"
            }
        },
        {
            "name": "angle",
            "label": "Show both sides at x =",
            "type": "number",
            "default": "60"
        }
    ],
    "outputs": [
        {
            "key": "verdict",
            "label": "Verdict",
            "format": "text"
        },
        {
            "key": "lhs_value",
            "label": "Left side at your angle",
            "format": "number"
        },
        {
            "key": "rhs_value",
            "label": "Right side at your angle",
            "format": "number"
        },
        {
            "key": "difference",
            "label": "Difference at your angle",
            "format": "number"
        },
        {
            "key": "max_difference",
            "label": "Largest relative difference found",
            "format": "text"
        },
        {
            "key": "points_checked",
            "label": "Angles compared",
            "format": "integer"
        },
        {
            "key": "counterexample",
            "label": "First counterexample",
            "format": "text"
        }
    ],
    "formula": {
        "text": "Evaluate L(x) and R(x) at 400 angles spread over two full turns (avoiding multiples of 90°); the identity fails if |L − R| > 10⁻⁹ × max(1, |L|, |R|) at any of them",
        "variables": {
            "L(x), R(x)": "the two sides you type",
            "x": "the angle, in degrees or radians as chosen"
        }
    },
    "worked_example": {
        "inputs": {
            "lhs": "cos(2x)",
            "rhs": "1 - 2sin^2(x)",
            "unit": "deg",
            "angle": "60"
        },
        "steps": [
            "At x = 60°: cos 120° = −0.5.",
            "Right side: 1 − 2 × sin²60° = 1 − 2 × 0.75 = −0.5.",
            "The same comparison at 400 angles finds no difference larger than floating-point rounding."
        ],
        "result": "cos 2x = 1 − 2 sin²x agreed at every one of the 400 test angles, as expected for this double-angle identity."
    },
    "interpretation": "A counterexample settles the question: the expressions are not identical. Agreement at every test angle is convincing evidence but not a proof; to prove an identity, transform one side into the other with known identities.",
    "assumptions": [
        "In degree mode, x and every trig function argument are in degrees, and pi stands for 180°.",
        "Angles where either side is undefined (for example tan at 90°) are skipped."
    ],
    "limitations": [
        "Numerical testing cannot prove an identity, and cannot detect differences smaller than about 10⁻⁹ of the values.",
        "Inverse trig functions and hyperbolic functions are not supported."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "How do I write sin squared?",
            "answer": "Type sin^2(x) or sin(x)^2. Write products with or without *: 2sin(x)cos(x) works."
        },
        {
            "question": "Why is this not a proof?",
            "answer": "Two different functions can agree at many points. The checker samples 400 angles, which catches every common mistake, but a formal proof uses algebraic steps."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Radians to Degrees Converter",
            "url": "https://calculators.geekmidst.com/calculators/radians-to-degrees-calculator"
        },
        {
            "name": "Law of Sines Calculator",
            "url": "https://calculators.geekmidst.com/calculators/triangle-law-of-sines-calculator"
        },
        {
            "name": "Triangle Calculator",
            "url": "https://calculators.geekmidst.com/calculators/triangle-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}