Sometimes we can think something is an identity when it actually isn’t. Consider the following equations that are sometimes mistaken as identities:

Both of these are true for some very specific values of and  (for example, when either or is 0), but they are not true for most values of and , for example and (try it!). The actual identities associated with the expressions on the left side are and .

Are polynomials the only types of expressions you can find in identities? Not at all! Here is an identity that shows a relationship between rational expressions:

We can show that this identity is true by adding the terms in the expression on the right using a common denominator:

An important difference from polynomial identities is that identities involving rational expressions could have a few exceptional values of for which they are not true because the rational expressions on one side or the other are not defined. For example, the identity above is true for all values of except and .