What is a trigonometric identity?

A trigonometric identity is an equation involving trigonometric functions (such as sin, cos, tan, sec, csc, cot) that is true for all values of the variable for which both sides are defined. Examples include sin^2 x + cos^2 x = 1 and tan x = sin x / cos x. Identities are used to simplify expressions, solve equations, and derive more advanced results in calculus, physics, and engineering.

How can I use this tool to check a trig identity?

Enter the left-hand side and right-hand side as functions of x using standard notation, for example sin(x)^2 + cos(x)^2 and 1. Then click "Check identity numerically". The tool evaluates both expressions at several values of x in radians and compares the results within a small tolerance. If they match at all test points, the identity is likely correct, but this is still a numerical check rather than a formal proof.

Which functions and constants does the trig identity checker understand?

The checker understands sin, cos, tan, sec, csc, cot, asin, acos, atan, sqrt, abs, log, ln, and exp. Use x as the variable, ^ for powers (e.g., sin(x)^2), and pi for the constant π. All trigonometric functions are evaluated in radians.

Are the numerical checks a proof that an identity is always true?

No. Numerical checks can strongly suggest that an identity is correct by testing many values of x, but they are not a formal proof. They can, however, quickly reveal mistakes in algebraic manipulations or help you gain confidence before writing a full symbolic proof on paper.

What are the most important trig identities to memorize?

At a minimum, you should know the Pythagorean identities, the reciprocal identities, the quotient identities, the even/odd identities, and the angle sum and difference formulas. Double-angle, half-angle, and product-to-sum identities are especially helpful in calculus and in simplifying integrals or Fourier series.

Trig Identity Explorer & Checker

Interactive trig identity explorer and cheat sheet. Browse core trigonometric identities and numerically verify your own identities with a trig identity checker.

Full original guide (expanded)

Trig Identity Explorer & Checker

Browse the most important trigonometric identities in one structured cheat sheet and use the interactive checker to test your own trig identities numerically.

All trigonometric functions are evaluated in radians. Use x as the variable.

Trig identity checker

Enter a candidate identity in terms of x. The tool evaluates both sides at several values of x (in radians) and checks whether they agree within a small numerical tolerance.

Left-hand side (LHS)

Right-hand side (RHS)

Supported functions: sin, cos, tan, sec, csc, cot, asin, acos, atan, sqrt, abs, log, ln, exp, and constant pi.

The numerical check report will appear here. The tool samples several values of x in radians and compares LHS(x) and RHS(x).

Basic trig identities

Definitions on the unit circle

For a real angle \(x\) measured in radians, on the unit circle:

\(\sin x = y\), the y–coordinate of the point on the unit circle
\(\cos x = x\), the x–coordinate of the point on the unit circle
\(\tan x = \dfrac{\sin x}{\cos x}\), whenever \(\cos x \neq 0\)

Reciprocal identities

\(\sec x = \dfrac{1}{\cos x}\), whenever \(\cos x \neq 0\)
\(\csc x = \dfrac{1}{\sin x}\), whenever \(\sin x \neq 0\)
\(\cot x = \dfrac{\cos x}{\sin x}\), whenever \(\sin x \neq 0\)

Pythagorean identities

\(\sin^2 x + \cos^2 x = 1\)
\(1 + \tan^2 x = \sec^2 x\), whenever \(\cos x \neq 0\)
\(1 + \cot^2 x = \csc^2 x\), whenever \(\sin x \neq 0\)

Quotient identities

\(\tan x = \dfrac{\sin x}{\cos x}\), whenever \(\cos x \neq 0\)
\(\cot x = \dfrac{\cos x}{\sin x}\), whenever \(\sin x \neq 0\)

Even–odd identities

\(\sin(-x) = -\sin x\)
\(\cos(-x) = \cos x\)
\(\tan(-x) = -\tan x\)
\(\csc(-x) = -\csc x\)
\(\sec(-x) = \sec x\)
\(\cot(-x) = -\cot x\)

Cofunction identities

For angles measured in radians:

\(\sin\left(\dfrac{\pi}{2} - x\right) = \cos x\)
\(\cos\left(\dfrac{\pi}{2} - x\right) = \sin x\)
\(\tan\left(\dfrac{\pi}{2} - x\right) = \cot x\)
\(\cot\left(\dfrac{\pi}{2} - x\right) = \tan x\)
\(\sec\left(\dfrac{\pi}{2} - x\right) = \csc x\)
\(\csc\left(\dfrac{\pi}{2} - x\right) = \sec x\)

How to read and use trig identities effectively

Trigonometric identities are more than just formulas to memorize: they encode deep geometric and analytic relationships between angles and triangles. When used well, they can turn complicated expressions into manageable ones and provide shortcuts in physics, engineering, and computer graphics.

1. Start from the Pythagorean identity

The identity \(\sin^2 x + \cos^2 x = 1\) is the foundation of most other trig identities. It follows directly from the unit circle definition and the Pythagorean theorem:

On the unit circle, a point has coordinates \((\cos x, \sin x)\). Using the Pythagorean theorem:

\(\cos^2 x + \sin^2 x = 1^2 = 1.\)

2. Verify identities numerically before proving them

The identity checker on this page is not a formal proof engine, but it is extremely useful as a sanity check. If two expressions are supposed to be identical but disagree numerically for some values of \(x\), then there is an algebraic mistake somewhere.

Typical workflow:

Manipulate the expression on paper using known identities.
Use the checker to compare the original expression and your simplified form for multiple values of \(x\).
If they agree numerically, you can be much more confident that your algebra is correct.

3. Watch out for domain issues

Many trig identities hold only where both sides are defined. For example, \(\tan x = \dfrac{\sin x}{\cos x}\) is undefined when \(\cos x = 0\). When checking identities numerically, the tool skips test points where either side is not a finite real number and reports any domain-related issues separately.

4. Connecting to calculus and complex analysis

In calculus, trig identities simplify derivatives and integrals, especially for periodic or oscillatory functions. In complex analysis, Euler’s formula \(e^{ix} = \cos x + i \sin x\) unifies many identities into simple algebraic relations between exponentials. While this tool focuses on real-variable identities, the same formulas extend naturally into the complex plane.

Trig identity – FAQ

Use the caret symbol ^ for powers. For example, write sin(x)^2 for \(\sin^2 x\), cos(x)^4 for \(\cos^4 x\), and so on. If you have a product like \(\sin^2 x \cos x\), you can type sin(x)^2 * cos(x).

Audit: Complete

Formula (LaTeX) + variables + units

This section shows the formulas used by the calculator engine, plus variable definitions and units.

Formula (extracted LaTeX)

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Formula (extracted text)

Definitions on the unit circle For a real angle \(x\) measured in radians, on the unit circle: \(\sin x = y\), the y–coordinate of the point on the unit circle \(\cos x = x\), the x–coordinate of the point on the unit circle \(\tan x = \dfrac{\sin x}{\cos x}\), whenever \(\cos x \neq 0\)

Formula (extracted text)

Reciprocal identities \(\sec x = \dfrac{1}{\cos x}\), whenever \(\cos x \neq 0\) \(\csc x = \dfrac{1}{\sin x}\), whenever \(\sin x \neq 0\) \(\cot x = \dfrac{\cos x}{\sin x}\), whenever \(\sin x \neq 0\)

Formula (extracted text)

Pythagorean identities \(\sin^2 x + \cos^2 x = 1\) \(1 + \tan^2 x = \sec^2 x\), whenever \(\cos x \neq 0\) \(1 + \cot^2 x = \csc^2 x\), whenever \(\sin x \neq 0\)

Formula (extracted text)

Quotient identities \(\tan x = \dfrac{\sin x}{\cos x}\), whenever \(\cos x \neq 0\) \(\cot x = \dfrac{\cos x}{\sin x}\), whenever \(\sin x \neq 0\)

Formula (extracted text)

Even–odd identities \(\sin(-x) = -\sin x\) \(\cos(-x) = \cos x\) \(\tan(-x) = -\tan x\) \(\csc(-x) = -\csc x\) \(\sec(-x) = \sec x\) \(\cot(-x) = -\cot x\)

Formula (extracted text)

Cofunction identities For angles measured in radians: \(\sin\left(\dfrac{\pi}{2} - x\right) = \cos x\) \(\cos\left(\dfrac{\pi}{2} - x\right) = \sin x\) \(\tan\left(\dfrac{\pi}{2} - x\right) = \cot x\) \(\cot\left(\dfrac{\pi}{2} - x\right) = \tan x\) \(\sec\left(\dfrac{\pi}{2} - x\right) = \csc x\) \(\csc\left(\dfrac{\pi}{2} - x\right) = \sec x\)

Variables and units

T = property tax (annual or monthly depending on input) (currency)

Sources (authoritative):

NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures
FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/

Changelog

Version: 0.1.0-draft
Last code update: 2026-01-19

0.1.0-draft · 2026-01-19

Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.

Verified by Ugo Candido on 2026-01-19
Profile · LinkedIn

All evaluations in the identity checker are done in radians, which is the standard in most higher mathematics and in calculus. You can use pi in your expressions: for instance, sin(pi/3) represents \(\sin( \pi / 3 )\), which equals \(\sqrt{3}/2\).

Core Math & Algebra tools

Trig & geometry tools

Professional use note

This trig identity explorer is ideal for study, exam preparation, and quick checks in technical work. For mission-critical engineering or scientific computations, always validate key transformations using independent methods or domain-specific software.

Formulas

(Formulas preserved from original page content, if present.)

Version 0.1.0-draft

Citations

Add authoritative sources relevant to this calculator (standards bodies, manuals, official docs).

Changelog

0.1.0-draft — 2026-01-19: Initial draft (review required).

Trig Identity Explorer & Checker

Full original guide (expanded)

Trig Identity Explorer & Checker

Trig identity checker

Basic trig identities

Sum and difference identities

Double-angle and half-angle identities

Product-to-sum and sum-to-product identities

How to read and use trig identities effectively

1. Start from the Pythagorean identity

2. Verify identities numerically before proving them

3. Watch out for domain issues

4. Connecting to calculus and complex analysis

Trig identity – FAQ