Pythagorean Identities — Formulas, Proof, Examples
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:
- sin²θ + cos²θ = 1 (I)
- 1 + tan²θ = sec²θ (II)
- 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.