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For some equation to be an identity, should the LHS and RHS have same domain (given equality holds for all values of x). For example, are both of these an identity: $$1.\ \sec^2(\theta)-\tan^2(\theta)=1$$$$2.\ \sec^2(\theta)=1+\tan^2(\theta)?$$

Are there any other conditions for the equation to be called an identity?

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1 Answer

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An equation is an identity when the left-hand side and the right-hand side agree at all points of the intersection of their domains. "Agree" means that arbitrary specialization of the variables appearing in the equation either produces an equality of values or is outside the domain of one or both sides.

If you additionally require that the domains of the two are equal, you are requiring equality of functions, which is a stricter requirement than identity.

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