What is an identity?

An identity is an equation that holds true for all values of the variables appearing in it, because it either is a definition or is the logical consequence of a definition. An example of a definitional identity is ${\displaystyle \tan(x)={\frac {\sin(x)}{\cos(x)}}.}$ An example of an identity that can logically be proven to hold for all values of its variable is the Pythagorean identity expressed in trigonometric form: ${\displaystyle sin^{2}(x)+cos^{2}(x)=1.}$

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