Identity (Equation)

High School

Definition

An identity is an equation that stays true no matter what values are substituted for any of its variables. It is an equality between functions that are defined differently. An example of this is tan θ = sin θ /cos θ. Both sides of this equation will produce the same values regardless of what θ¸ ends up becoming.

Worked examples

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

This Pythagorean identity is true for every angle—substitute any θ and both sides match.

\(a + 0 = a\)

The additive identity holds for all real numbers a; zero never changes the value.

\((x + 1)^2 = x^2 + 2x + 1\)

Expanding the left side always equals the right, no matter what x is—both are the same function.

Common mistakes

\(\sin \theta + \cos \theta = 1\) → \(\sin^2 \theta + \cos^2 \theta = 1\) The Pythagorean identity requires squaring each function; the sum of the functions themselves is not constant.
thinking \(x^2 = 9\) is an identity → it is a conditional equation (only true when \(x = \pm 3\)) An identity must be true for all values, not just specific solutions.
\(\tan \theta = \frac{\sin \theta}{\cos \theta}\) only when \(\theta = 45^\circ\) → the identity holds for all \(\theta\) where \(\cos \theta \ne 0\) An identity is universally true (within its domain), not just for one special angle.

Where you'll use it next

You'll use identities to simplify and prove trigonometric and algebraic equations in pre-calculus, verify solutions in calculus, and solve systems in linear algebra and differential equations.

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Pythagorean identities

Grade 12 Math

Pythagorean identities are formulas, derived from Pythagorean Theorem, that allow us to find out where a point is on the unit circle. Learn the tricks and tips on how to use the unit circle to derive and prove the Pythagorean identities can be difficult.

12th Grade12th