Identity

Q: What are the most common identities in math?

In algebra, the most used identities include $(a + b)^2 = a^2 + 2ab + b^2$, $(a - b)^2 = a^2 - 2ab + b^2$, and $a^2 - b^2 = (a-b)(a+b)$. In trigonometry, the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ is fundamental, along with the double-angle and sum/difference identities.

Identity (Equation or Inequality)

An equation which is true regardless of what values are substituted for any variables (if there are any variables at all).

Identities:

1 + 1 = 2

(x + y)² = x² + 2xy + y²

a² ≥ 0

sin² θ + cos² θ = 1

Key Formula

\text{An identity holds for all permissible values of its variables.}

\text{Examples: } (a + b)^2 = a^2 + 2ab + b^2, \quad \sin^2\theta + \cos^2\theta = 1

Where:

$a, b$ = Any real numbers (or variables representing real numbers)
$\theta$ = Any angle measure (in degrees or radians)

Worked Example

Problem: Verify that

(x + 3)^2 = x^2 + 6x + 9

is an identity by expanding the left side, then confirm with two different values of

x

Step 1: Expand the left side using the rule

(a + b)^2 = a^2 + 2ab + b^2

with

a = x

and

b = 3

(x + 3)^2 = x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9

Step 2: The expanded form matches the right side exactly, so this equation is an identity — it must be true for every value of

x

x^2 + 6x + 9 = x^2 + 6x + 9 \quad \checkmark

Step 3: Confirm with

x = 2

: the left side is

(2 + 3)^2 = 25

, and the right side is

4 + 12 + 9 = 25

(2 + 3)^2 = 25 \quad\text{and}\quad 2^2 + 6(2) + 9 = 25

Step 4: Confirm with

x = -5

: the left side is

(-5 + 3)^2 = 4

, and the right side is

25 - 30 + 9 = 4

(-5 + 3)^2 = 4 \quad\text{and}\quad (-5)^2 + 6(-5) + 9 = 4

Answer: Both sides simplify to the same expression, confirming

(x + 3)^2 = x^2 + 6x + 9

is an identity.

Another Example

This example highlights an important subtlety: an identity must hold for all permissible (domain-valid) values, not literally every real number. A restricted domain does not disqualify an equation from being an identity.

Problem: Determine whether the equation

\dfrac{x^2 - 4}{x - 2} = x + 2

is an identity.

Step 1: Factor the numerator on the left side using the difference of squares.

\frac{x^2 - 4}{x - 2} = \frac{(x - 2)(x + 2)}{x - 2}

Step 2: Cancel the common factor

(x - 2)

, noting this is valid only when

x \neq 2

\frac{(x - 2)(x + 2)}{x - 2} = x + 2 \quad (x \neq 2)

Step 3: The left side is undefined at

x = 2

because the denominator equals zero, while the right side equals

4

x = 2

. Both sides agree for every permissible value of

x

(i.e., every

x

in the domain of the left side).

\text{Domain: } x \in \mathbb{R},\; x \neq 2

Answer: Yes, this is an identity on its natural domain. For every value of

x

where both sides are defined (

x \neq 2

), the equation holds true.

Frequently Asked Questions

What is the difference between an identity and an equation?

Every identity is an equation, but not every equation is an identity. An identity is true for all permissible values of its variables, like

a^2 - b^2 = (a-b)(a+b)

. A conditional equation, by contrast, is true only for certain values — for example,

2x + 1 = 7

is true only when

x = 3

How do you prove something is an identity?

You prove an identity by algebraically transforming one side until it matches the other, or by simplifying both sides to the same expression. Plugging in specific numbers can help you check your work or disprove a supposed identity, but substitution alone cannot prove one — because you would need to test infinitely many values.

What are the most common identities in math?

In algebra, the most used identities include

(a + b)^2 = a^2 + 2ab + b^2

(a - b)^2 = a^2 - 2ab + b^2

, and

a^2 - b^2 = (a-b)(a+b)

. In trigonometry, the Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

is fundamental, along with the double-angle and sum/difference identities.

Identity vs. Conditional Equation

	Identity	Conditional Equation
Definition	An equation true for all permissible values of its variables	An equation true only for specific values of its variables
Example	$(x+1)^2 = x^2 + 2x + 1$	$x^2 + 2x + 1 = 16$
Number of solutions	All values in the domain satisfy it	A finite number of values (or sometimes none)
Goal	Prove or verify that both sides are equivalent expressions	Solve for the specific value(s) that make it true
Notation hint	Sometimes written with ≡ to emphasize it is an identity	Written with the standard = sign

Why It Matters

Identities are the backbone of algebraic simplification — every time you factor, expand, or cancel terms, you are relying on an identity. In trigonometry courses, proving and applying trig identities is a major topic, and these skills carry directly into calculus when you simplify integrals and derivatives. Recognizing whether a statement is an identity or a conditional equation also determines whether you should simplify or solve, which is a foundational skill in all of mathematics.

Common Mistakes

Mistake: Testing a few values and concluding the equation must be an identity.

Correction: Numerical checks can disprove an identity (one counterexample is enough) but cannot prove one. You must show algebraically that both sides are equivalent expressions.

Mistake: Treating an equation like an identity when it is actually conditional, and simplifying instead of solving.

Correction: Before manipulating an equation, determine whether your goal is to prove equivalence (identity) or to find specific solutions (conditional equation). If the problem says 'solve,' you need particular values; if it says 'verify' or 'prove,' you need an algebraic argument.

Related Terms

Equation — General term; identities are a special type of equation
Variable — Identities hold for all values of their variables
Conditional Equation — An equation true only for certain variable values
Conditional Inequality — An inequality true only for certain variable values
Trig Identities — Important identities involving trigonometric functions
Difference of Squares — The algebraic identity $a^2 - b^2 = (a-b)(a+b)$
Perfect Square Trinomial — Arises from the identity $(a+b)^2 = a^2+2ab+b^2$