Trig identity verifier

The trig identity verifier turns a single finite angle into a numerical stress test of the identities every student and engineer relies on. At that angle it evaluates sine, cosine, tangent, cotangent, secant, and cosecant, then measures how far five classic statements drift from exact equality: the three Pythagorean forms sin squared plus cos squared equals one, one plus tan squared equals sec squared, and one plus cot squared equals csc squared, plus the two quotient forms tan equals sin over cos and cot equals cos over sin. Each identity returns left and right sides, an absolute residual, and a holds flag against your tolerance. The path is fully deterministic: no network, no randomness, no clock.

Why Pythagorean and quotient identities matter in practice

Trigonometric identities are not decorative textbook theorems; they are the algebraic backbone of unit-circle geometry, Fourier analysis, rotation matrices, and every library that rewrites expressions before differentiating or simplifying them. The fundamental Pythagorean identity sin squared theta plus cos squared theta equals one is the statement that a point on the unit circle has radius one. From it, dividing by cos squared or sin squared yields the companion forms that relate tangent to secant and cotangent to cosecant.

The quotient identities tan equals sin over cos and cot equals cos over sin are the definitions that connect ratio language to coordinate language. Students meet them when they simplify homework expressions; engineers meet them when a CAD or signal-processing pipeline rewrites an expression and must stay within floating-point tolerance of the original. Numerically checking residual error at a concrete angle is the fastest way to catch a mode error, a degree-versus-radian bug, or a hand-derived formula that almost holds but fails near a singularity.

This tool focuses on that verification contract: one angle in, five residual reports out, with an honest failure when any of the six core functions is undefined. It does not attempt symbolic proof; it answers the practical question of how large the floating-point residual is right here, which is what automated tests and grading scripts actually need.

Where identity verification fits in study, teaching, and automation

Students use residual checks when they want confidence that a calculator and a textbook identity agree at thirty degrees, at pi over four, or at an awkward angle their homework invented. Teachers can generate live demos that flip the angle and show every residual update in the browser without trusting opaque apps that stay stuck in the wrong unit mode. Authors of property-based tests can sample many angles, skip or expect failures at the axes where functions are undefined, and assert that max residual stays below a chosen epsilon.

Game and robotics developers who store orientations in degrees but feed radians into engines can verify that intermediate trig rewrites still obey Pythagorean structure within machine precision. Survey and CAD pipelines that rewrite elevation formulas benefit from a cheap gate that rejects undefined angles instead of propagating Infinity into a mesh. Pair it with degree–radian conversion when you need unit labels, with unit-circle helpers when you need exact textbook points, and with inverse trigonometric tools when you start from a ratio instead of an angle.