Trigonometric identity

A trigonometric identity is an equation involving trigonometric functions that holds for every value in its domain. It expresses a relationship that remains true across the functions’ allowed inputs.

Sine: A trigonometric function that assigns a ratio of sides to an angle in a right triangle and extends periodically to real inputs. Its periodic values appear throughout identities, often paired with cosine.
Pythagorean trigonometric identity: The identity sin²x + cos²x = 1 and its forms obtained by division. It is the central relation from which several standard identities follow.
Trigonometric equation: An equation involving trigonometric functions whose solutions are sought for particular input values. Unlike an identity, it need only hold for its solution values.
Trigonometric inequality: An inequality involving trigonometric functions that is true for specified inputs or ranges. It asserts an order relation rather than equality at every allowed input.
Cosine: A trigonometric function that assigns a ratio of sides to an angle in a right triangle and extends periodically to real inputs. It completes the fundamental relation with sine: their squared values sum to one.
Angle addition formulas: Formulas expressing trigonometric functions of sums or differences of angles using functions of each angle. They generate many identities and reveal how angle combinations behave.
Trigonometric simplification: The rewriting of trigonometric expressions into equivalent forms that are shorter or more useful. Identities provide the valid transformations used in simplification.
Conditional equation: An equation that is true only for some values of its variables. A trigonometric identity is universal on its domain, not limited to selected solutions.
Tangent: A trigonometric function equal to sine divided by cosine wherever cosine is nonzero. Its quotient definition lets many identities be derived from sine and cosine.
Double-angle formulas: Identities expressing trigonometric functions of twice an angle in terms of functions of the original angle. They are direct consequences of the angle addition formulas.

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