Identity (mathematics)

In mathematics an identity is an equality relation A = B, such that A and B contain some variables and A and B produce the same value as each other regardless of what values (usually numbers) are substituted for the variables. In other words, A = B is an identity if A and B define the same functions. This means that an identity is an equality between functions that are differently defined. For example, (a + b)² = a² + 2ab + b² and cos²(x) + sin²(x) = 1 are identities. Identities are sometimes indicated by the triple bar symbol $\equiv$ instead of $=$ , the equals sign.

Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities involving both angles and side lengths of a triangle. Only the former are covered in this article.

These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

One example is $\sin ^{2}\theta +\cos ^{2}\theta \equiv 1,\,$ $\sin ^{2}\theta +\cos ^{2}\theta \equiv 1,\,$ which is true for all complex values of $\theta$ $\theta$ (since the complex numbers ${\mathbb {C}}$ ${\mathbb {C}}$ are the domain of sin and cos), as opposed to

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