Identity (mathematics)

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an identity is an <a href="/facts/Equality_(mathematics)/AQCuumLg">equality</a> relating one <a href="/facts/Mathematical_expression/MPrMlYbE">mathematical expression</a> A to another mathematical expression B, such that A and B (which might contain some <a href="/facts/Variable_(mathematics)/VbQlrvKz">variables</a>) produce the same value for all values of the variables within a certain <a href="/facts/Domain_of_discourse/T1WRQxx6">domain of discourse</a>. In other words, A = B is an identity if A and B define the same <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a>, and an identity is an equality between functions that are differently defined. For example, 
 
 
 
 (
 a
 +
 b
 
 )
 
 2
 
 
 =
 
 a
 
 2
 
 
 +
 2
 a
 b
 +
 
 b
 
 2
 
 
 
 
 {\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}}
 
 and 
 
 
 
 
 cos
 
 2
 
 
 ⁡
 θ
 +
 
 sin
 
 2
 
 
 ⁡
 θ
 =
 1
 
 
 {\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1}
 
 are identities. Identities are sometimes indicated by the <a href="/facts/Triple_bar/txReSBs4">triple bar</a> symbol ≡ instead of =, the <a href="/facts/Equals_sign/AweYSg9h">equals sign</a>. Formally, an identity is a <a href="/facts/Universally_quantified/agNmvJKN">universally quantified</a> equality.

Identity (mathematics) open-in-new

Identity (mathematics)