Trigonometric Identities: The Equations That Are Always True

Most equations are true only for certain values. A trigonometric identity is special: it's true for every angle, no exceptions. That makes identities powerful tools — you can substitute one side for the other anytime to simplify a complicated expression or to find a missing trig value. A short list of identities, led by the famous Pythagorean identity, covers what the ACT and an introductory course expect.

This guide presents the core identities and how to use them, building on SOH-CAH-TOA and the unit circle, with worked and practice problems from Northside Tutoring.

Why Identities Matter

Identities let you rewrite trig expressions into simpler equivalent forms and connect the trig functions to one another. On the ACT they show up in simplification and "find the value" problems; in Pre-Calculus they're essential for solving trig equations. (Beyond the SSAT; the SAT tests them lightly at most.)

What an Identity Is

An identity is an equation that holds for all valid inputs. Unlike an equation you solve for a specific x, an identity is always true — so you can swap one expression for an equal one whenever it's convenient.

The Pythagorean Identity

The single most important trig identity comes straight from the unit circle (where a point is (cosθ, sinθ) on a circle of radius 1):

This lets you find cosine from sine or vice versa. If sinθ = 3/5, then cos²θ = 1 − 9/25 = 16/25, so cosθ = ±4/5 (sign depends on the quadrant).

The Quotient Identity

Tangent is sine over cosine:

So once you know sine and cosine, you know tangent — just divide.

Reciprocal Identities

The three less-common functions are just reciprocals of the familiar ones:

Function	Equals
cosecant (csc)	1 / sin
secant (sec)	1 / cos
cotangent (cot)	1 / tan

Using Identities

Two main uses. First, simplify: replace a complicated expression with an equal, simpler one (for example, 1 − sin²θ becomes cos²θ). Second, find a value: given one function's value, use the Pythagorean and quotient identities to find the others.

Where You'll See This — Test by Test

No reference sheet provides identities — memorize the Pythagorean identity especially. The ACT tests them in simplification and value problems; they're core Pre-Calculus and beyond the SSAT.

Watch the Lesson

Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.

Worked Example Problems

These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.

Common Mistakes to Avoid

Practice Problems — You Try

Three problems below. Work each before checking the solution.

The Northside Method — How We Teach This 1-on-1

Reading a blog is a great starting point. But there's a meaningful gap between understanding a concept and reflexively applying it under timed conditions. That gap is exactly what our tutors close.

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