A trigonometric identity is an equation involving trigonometric functions, sine, cosine, tangent and their relatives, that holds true for literally every possible value of the angle involved, not just for one specific angle or a limited set of them. This is what genuinely distinguishes an identity from an ordinary trigonometric equation: solving an ordinary equation means finding the specific angle or angles that make it true, while verifying or applying an identity means recognising a relationship that's already true universally, for any angle at all, and using that universal relationship to simplify or rewrite an expression.
The Pythagorean identity holds regardless of which angle is chosen
The most fundamental trigonometric identity, sin squared theta plus cos squared theta equals one, follows directly from the Pythagorean theorem applied to a right triangle inscribed in a unit circle, and it holds true for every possible value of theta, not merely for some angles and not others. Because a point on the unit circle at angle theta always has coordinates whose horizontal and vertical components are, by definition, cosine theta and sine theta respectively, and because that point always sits exactly one unit from the circle's centre by the circle's own definition, the Pythagorean relationship between those two coordinates is guaranteed to hold no matter which specific angle theta actually is.
Other identities let an expression be rewritten without changing its actual value
Beyond the Pythagorean identity, trigonometry includes a substantial family of other identities, sum and difference formulas, double-angle formulas, reciprocal and quotient identities, each expressing some genuinely universal relationship among trigonometric functions that holds for every angle. Because these relationships are always true, an identity can be used to rewrite a trigonometric expression into an equivalent but often more useful or simpler form without ever changing the expression's actual underlying value for any given angle — this is precisely why identities are such a heavily used tool throughout trigonometry and calculus, letting a genuinely difficult expression be transformed into a more tractable equivalent one.
What we're still unsure about
The Pythagorean identity's derivation, and the broader distinction between an identity and an ordinary solvable equation, are precisely defined, fully settled mathematics with no genuine ambiguity. What varies is more a matter of pedagogy than of open mathematical questions: different maths curricula and textbooks vary noticeably in which specific identities they expect a student to memorise outright versus derive on the fly from a smaller core set, without full agreement across educators on exactly where that particular line should be drawn for a given course level.
This sits inside Trigonometric Identities, one of seven topics in Trigonometry, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.