# Trig Identities Calculator

URL: https://speedycalculator.com/calculators/math/trig-identities/
Last updated: 2026-10-30

> The trig identities calculator checks the main families of trigonometric identities – Pythagorean, reflection, shift, double, triple and half angle – numerically for any angle θ.

## Inputs

- Angle unit
- Angle θ
- Identity type

## Results

- Identities at θ
- sin θ
- cos θ
- tan θ

## Formula

```
sin²θ + cos²θ = 1,  1 + tan²θ = sec²θ,  1 + cot²θ = csc²θ
sin(−θ) = −sin θ,  cos(θ + π/2) = −sin θ,  sin 3θ = 3 sin θ − 4 sin³θ …
```

## Explanation

Trigonometric identities are equations true for every angle. They are the toolbox for simplifying expressions, solving trig equations and integrating. Seeing them evaluated at a specific angle is a quick way to remember and check them.

### How to use the Trig Identities Calculator

- **Angle unit**
- **Angle θ**
- **Identity type**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Pick the angle and a family of identities. The calculator evaluates both sides of each identity in that family at your angle.

## Worked example

Inputs — Angle unit: Degrees (°); Angle θ: 30; Identity type: Pythagorean.

Results — **Identities at θ = sin²θ + cos²θ = 0.25 + 0.75 = 1
1 + tan²θ = sec²θ = 1.333333333
1 + cot²θ = csc²θ = 4**; sin θ = 0.5; cos θ = 0.8660254038; tan θ = 0.5773502692.

## Key terms

- **Identity**: An equation true for all values of the variable.

## FAQ

**What is the most important trig identity?**
sin²θ + cos²θ = 1, the Pythagorean identity.

**Are identities the same as equations?**
No – an identity holds for every angle, an equation only for some.

**How do I prove an identity?**
Transform one side into the other using known identities.

**Which functions are odd?**
Sine, tangent, cotangent and cosecant; cosine and secant are even.


## Sources

- [OpenStax – Algebra and Trigonometry 2e](https://openstax.org/details/books/algebra-and-trigonometry-2e)
- [OpenStax – Precalculus 2e](https://openstax.org/details/books/precalculus-2e)
- [Wolfram MathWorld – Trigonometric identities](https://mathworld.wolfram.com/TrigonometryIdentities.html)