# Pythagorean Identities — Formulas, Proof, Examples

The three Pythagorean identities — sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ — are the load-bearing equations of trigonometry. All three come from the geometric fact: every point on the unit circle satisfies x² + y² = 1. Once you know the first, the other two follow by dividing through by cos²θ and sin²θ.

## A Theorem That Showed Up Two Thousand Years Late

Pythagoras proved a² + b² = c² for right triangles around 530 BCE. Sine and cosine — measured as ratios in those same right triangles — weren't given systematic names until Aryabhata's _Aryabhatiya_ in 499 CE. When the names finally arrived, the identity sin²θ + cos²θ = 1 wasn't a new theorem. It was the _Pythagorean theorem_ itself, dressed in new clothes.

## What Are the Pythagorean Identities?

The **Pythagorean identities** are three trigonometric identities that follow directly from applying the Pythagorean theorem to the unit circle. They hold for every angle θ where the functions involved are defined.

The three are:

1. sin²θ + cos²θ = 1 (I)
2. 1 + tan²θ = sec²θ (II)
3. 1 + cot²θ = csc²θ (III)

Identity (I) is the master; (II) and (III) are derived from it in two algebraic steps each.

## Proof — Unit Circle Method

Place a point P on the unit circle so that the line from the origin to P makes angle θ with the positive x-axis.

By definition: P = (cosθ, sinθ).

The unit circle satisfies x² + y² = 1 for every point on it. Substituting:

cos²θ + sin²θ = 1

That's identity (I).

### Deriving identities (II) and (III)

**Divide (I) by cos²θ (allowed when cosθ ≠ 0):**
sin²θ/cos²θ + cos²θ/cos²θ = 1/cos²θ

tan²θ + 1 = sec²θ

That's identity (II).

**Divide (I) by sin²θ (allowed when sinθ ≠ 0):**
sin²θ/sin²θ + cos²θ/sin²θ = 1/sin²θ

1 + cot²θ = csc²θ

That's identity (III).

## Proof — Right Triangle Method

For an acute angle θ in a right triangle with legs a, b and hypotenuse c:

sinθ = a/c, cosθ = b/c

By Pythagoras: a² + b² = c². Divide both sides by c²:

a²/c² + b²/c² = 1 ⟹ sin²θ + cos²θ = 1

Same identity, same proof line, restricted to acute angles. The unit-circle proof generalizes this to every real θ, including negatives and angles past π/2.

## The Three Forms of Each Identity

Each Pythagorean identity has three useful rearrangements:

| Identity | Form A | Form B | Form C |
| --- | --- | --- | --- |
| (I) | sin²θ + cos²θ = 1 | sin²θ = 1 - cos²θ | cos²θ = 1 - sin²θ |
| (II) | 1 + tan²θ = sec²θ | tan²θ = sec²θ - 1 | sec²θ - tan²θ = 1 |
| (III) | 1 + cot²θ = csc²θ | cot²θ = csc²θ - 1 | csc²θ - cot²θ = 1 |

## Why the Pythagorean Identities Are Worth Knowing Cold

These identities are not just exam furniture. They are the backbone of every trig manipulation done in physics, signal processing, and engineering.

- **Simple harmonic motion.** Total energy is constant due to sin² + cos² = 1.
- **Signal processing.** The factor of 1/2 comes from time-averaging sin²θ.
- **GPS and astronomy.** Earth position calculations involve sin²(latitude) + cos²(latitude) = 1.

## Key Takeaways
- The three **Pythagorean identities** are sin² + cos² = 1, 1 + tan² = sec², and 1 + cot² = csc².
- Identity (I) is the unit-circle equation in trigonometric terms.
- Identities (II) and (III) are one division each away from (I).
- The biggest exam slip is choosing the wrong sign after taking a square root.