An identity is an equation that is true for every value of its variables, like the Special Products $(a + b)^2 = a^2 + 2ab + b^2$. Identities let you rewrite expressions, do mental arithmetic and even build things, using the structure you have been practicing.
Key idea
To prove an identity, simplify one side until it matches the other (or both sides to the same thing). To disprove one, a single counterexample is enough. A useful example: $(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2$ produces Pythagorean triples.
Why it works
Expanding uses only the rules of arithmetic, which hold for every number, so a matching expansion proves the equation for every value. For the triple identity, expand both sides: the left is $x^4 + 2x^2y^2 + y^4$, and the right is $x^4 - 2x^2y^2 + y^4 + 4x^2y^2$, which is the same. So any whole numbers $x > y$ give whole numbers $a = x^2 - y^2$, $b = 2xy$, $c = x^2 + y^2$ with $a^2 + b^2 = c^2$.
| $x$ | $y$ | $x^2 - y^2$ | $2xy$ | $x^2 + y^2$ |
|---|---|---|---|---|
| 2 | 1 | 3 | 4 | 5 |
| 3 | 2 | 5 | 12 | 13 |
| 4 | 1 | 15 | 8 | 17 |
| 4 | 3 | 7 | 24 | 25 |
Worked examples
Worked example: Proving an identity
Show that $(x + 1)^3 - (x - 1)^3 = 6x^2 + 2$.
- $(x + 1)^3 = x^3 + 3x^2 + 3x + 1$ and $(x - 1)^3 = x^3 - 3x^2 + 3x - 1$.
- Subtract: the $x^3$ and $3x$ terms cancel, leaving $6x^2 + 2$.
Answer: Both sides are $6x^2 + 2$ for every $x$.
Worked example: Mental arithmetic
Find $102^2$ and $49 \times 51$ without a calculator.
- $102^2 = (100 + 2)^2 = 10000 + 400 + 4$ $= 10404$.
- $49 \times 51 = (50 - 1)(50 + 1) = 2500 - 1$ $= 2499$.
Answer: $10404$ and $2499$
Common mistake
Believing $(x + y)^2 = x^2 + y^2$. It fails at $x = y = 1$: the left side is 4 and the right side is 2. The missing $2xy$ is the two rectangles in the area picture. One counterexample is enough to show an equation is not an identity.