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What is the difference between an identity, an equation and a conditional equation?

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up vote 2 down vote accepted

Equation means equality. They are both related to the word equal. If such an equality is true for all values of the variable, it is called an identity, e.g., $\sin^2x+\cos^2x=1$ is true for all x. If however the equation in question only holds for some values, which one is supposed to determine, then it's called conditional, and its variable is termed an unknown.

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An identity may also contain no variables, such as $e^{i\pi}+1=0$. – zz20s yesterday
    
Sometimes identities have restrictions on the allowed values. Maybe that's poor form on the part of textbook authors though. E.g. $\cos x \tan x = \sin x$ is usually called an identity even though the left and right sides could be argued to have distinct domains. Of course, this is going by the language in a precalc book. – jdods yesterday

Equation is a mathematical description which is equal only for one variable. But identity is a mathematical description which is always equal for any variable.

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