# Trigonometric Identity Checker

> Type both sides of a trigonometric identity and test them numerically at 400 angles to see whether they agree, with a counterexample if they do not.

Category: Algebra, geometry & advanced math · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/trigonometric-identity-verification-calculator

Enter the left and right sides as expressions in x. The checker evaluates both at hundreds of angles: a single mismatch proves the identity false, while agreement everywhere is strong evidence (not a proof) that it is true.

Also called: Trig identity verifier.

## Inputs
- Left side (default cos(2x))
- Right side (default 1 - 2sin^2(x))
- Angles in (options: Degrees, Radians; default deg)
- Show both sides at x = (default 60)

## Results
- Verdict
- Left side at your angle
- Right side at your angle
- Difference at your angle
- Largest relative difference found
- Angles compared
- First counterexample

## Formula

```
Evaluate L(x) and R(x) at 400 angles spread over two full turns (avoiding multiples of 90°); the identity fails if |L − R| > 10⁻⁹ × max(1, |L|, |R|) at any of them
```

Where:
- L(x), R(x): the two sides you type
- x: the angle, in degrees or radians as chosen

## Worked example
1. At x = 60°: cos 120° = −0.5.
2. Right side: 1 − 2 × sin²60° = 1 − 2 × 0.75 = −0.5.
3. The same comparison at 400 angles finds no difference larger than floating-point rounding.

**Result:** cos 2x = 1 − 2 sin²x agreed at every one of the 400 test angles, as expected for this double-angle identity.

## What the result means

A counterexample settles the question: the expressions are not identical. Agreement at every test angle is convincing evidence but not a proof; to prove an identity, transform one side into the other with known identities.

## Assumptions and limits
- In degree mode, x and every trig function argument are in degrees, and pi stands for 180°.
- Angles where either side is undefined (for example tan at 90°) are skipped.
- Numerical testing cannot prove an identity, and cannot detect differences smaller than about 10⁻⁹ of the values.
- Inverse trig functions and hyperbolic functions are not supported.

## Questions

### How do I write sin squared?

Type sin^2(x) or sin(x)^2. Write products with or without *: 2sin(x)cos(x) works.

### Why is this not a proof?

Two different functions can agree at many points. The checker samples 400 angles, which catches every common mistake, but a formal proof uses algebraic steps.

## Related calculators
- [Radians to Degrees Converter](https://calculators.geekmidst.com/calculators/radians-to-degrees-calculator)
- [Law of Sines Calculator](https://calculators.geekmidst.com/calculators/triangle-law-of-sines-calculator)
- [Triangle Calculator](https://calculators.geekmidst.com/calculators/triangle-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/trigonometric-identity-verification-calculator