---
title: "The Pythagorean and Quotient Identities"
description: "sin²θ + cos²θ = 1 is Pythagoras on the unit circle, and tan θ = sin θ / cos θ is a ratio of coordinates. Use both to simplify expressions and to prove further identities."
canonical: https://lightmysky.com/learn/mathematics/the-pythagorean-and-quotient-identities-mt_NXBq37vxQh
source: https://lightmysky.com/learn/mathematics/the-pythagorean-and-quotient-identities-mt_NXBq37vxQh.md
retrieved: 2026-09-12
---

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# The Pythagorean and Quotient Identities

sin²θ + cos²θ = 1 is Pythagoras on the unit circle, and tan θ = sin θ / cos θ is a ratio of coordinates. Use both to simplify expressions and to prove further identities.

Subject: Mathematics · Area: Geometry · Ages 16 to 17
Page: https://lightmysky.com/learn/mathematics/the-pythagorean-and-quotient-identities-mt_NXBq37vxQh

## Ready when they can

- Derive sin²θ + cos²θ = 1 from the radius of the unit circle
- Simplify (1 - cos²θ) / sin θ to a single function
- Prove a given identity by working on one side until it matches the other

## Lesson: Two identities from one circle

Every point on the unit circle obeys x squared plus y squared equals 1. With x as cosine and y as sine, that reads sin squared plus cos squared equals 1. Pythagoras on radius 1 proves it for every angle.

Tangent is a ratio of coordinates. Tan theta equals sin theta over cos theta. Where cosine is zero the ratio has no value. That single line turns many hard forms into familiar ones.

**Example.** Simplify 1 minus cos squared theta, all over sin theta. The top matches the identity, so it becomes sin squared theta. One sine cancels with the bottom. The whole form collapses to sin theta.

**Tip.** To verify an identity, work on one side until it matches the other. Never move terms across the equals sign. Start with the messier side and quote each step. Given sin theta is 0.6 acute, cos squared is 1 minus 0.36, so cos theta is 0.8.

**Recap.** Read both identities off the circle, then use them to rewrite one side at a time.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [Pythagoras' Theorem](https://lightmysky.com/learn/mathematics/pythagoras-theorem-mt_1VmTUxBrNd)
- [The Unit Circle and the Graphs of Sine, Cosine and Tangent](https://lightmysky.com/learn/mathematics/the-unit-circle-and-the-graphs-of-sine-cosine-and-tangent-mt_VJaimevi3-)

## Opens up

- [Solving Trigonometric Equations in a Given Interval](https://lightmysky.com/learn/mathematics/solving-trigonometric-equations-in-a-given-interval-mt_74_jiK2xXA)
- [Parametric Equations of Curves](https://lightmysky.com/learn/mathematics/parametric-equations-of-curves-mt_7NC7SCeU2P)
- [Double Angle Formulae](https://lightmysky.com/learn/mathematics/double-angle-formulae-mt_AC9MmZ_fam)
- [Hyperbolic Functions and Their Inverses](https://lightmysky.com/learn/mathematics/hyperbolic-functions-and-their-inverses-mt_gCUl1sWi9M)
- [Trigonometric Integrals and Trigonometric Substitution](https://lightmysky.com/learn/mathematics/trigonometric-integrals-and-trigonometric-substitution-mt_JqysS41psg)