Consider the following relation:

\({\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\)

This is true for every value of *a* and *b*, and hence this is an example of an **identity**. In simple words, identities are equalities which are *always* true. Here is another example of an identity (which is related to the one above):

\[{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\]

**Example 1:** Evaluate \({1001^2}\) using the first identity above.

**Solution:** We have

\[\begin{align}&{1001^2} = {\left( {1000 + 1} \right)^2}\\&\;\;\;\;\;\;\;\;\; = {1000^2} + 2\left( {1000} \right)\left( 1 \right) + {1^2}\\&\;\;\;\;\;\;\;\;\;= 1000000 + 2000 + 1\\&\;\;\;\;\;\;\;\;\;= 1002001\end{align}\]

**Example 2:** Evaluate \({998^2}\)using the second identity above.

**Solution:** We have

\[\begin{array}{l}{998^2} = {\left( {1000 - 2} \right)^2}\\\;\;\;\;\;\;\;\; = {1000^2} - 2\left( {1000} \right)\left( 2 \right) + {2^2}\\\;\;\;\;\;\;\;\; = 1000000 - 4000 + 4\\\;\;\;\;\;\;\;\; = 996004\end{array}\]