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An identity is true for all values of the variable whereas an equation is true for only a finite number of values.

For example,

Identity: (x + 2)3 = x3 + 6x2 + 12x + 27 is true, whatever the value of x.

But

Equation: x3 - x = 0 is true only when x = -1, 0 or 1.

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Q: How would you explain to someone who has not yet studied trigonometry the difference between an identity and an equation?
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