Pythagorean identities · trigonometric identity
sin²θ + cos²θ = 1 — Pythagorean identity
When to use it
Two squared trig terms of the same angle are added: the pair is exactly 1, so replace both with 1 — or, going the other way, replace a stubborn 1 with sin²θ + cos²θ to get a common denominator.
Why it is true
A point on the unit circle at angle θ has coordinates (cos θ, sin θ). Its distance from the origin is the radius, 1. Pythagoras on those coordinates is exactly sin²θ + cos²θ = 1.
The full line-by-line version is on the proof page for pythagorean identity; the “how would I find this myself” version is in the derivation.
Where it comes from
From the definition of sine and cosine. You do not need to memorise this one — it is the Pythagorean theorem written in trigonometric notation. If you ever feel it is 'magic', the radius of the circle is the missing ingredient.
- x² + y² = r²Equation of a circle centred at the origin.
- x = cos θ, y = sin θ, r = 1Unit-circle definitions.
- cos²θ + sin²θ = 1Substitute and simplify.
Worked examples
- Simplify sin²θ + cos²θ + 3cos²θ1 + 3cos²θ — and with the power-reducing formula, 5/2 + (3/2)cos 2θ.
- Verify tan θ + cot θ = sec θ csc θConvert to sin and cos, take the common denominator, and the Pythagorean identity closes it.
- Why does cos 2x have three formulas and how do you choose?They are one formula plus the Pythagorean identity; choose by which square is already in your expression.
Checked at these angles
Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.
| Angle | Left side | Right side | Agree |
|---|---|---|---|
| 0° | 1 | 1 | yes |
| 7.5° | 1 | 1 | yes |
| 15° | 1 | 1 | yes |
| 18° | 1 | 1 | yes |
| 22.5° | 1 | 1 | yes |
| 30° | 1 | 1 | yes |
| 37° | 1 | 1 | yes |
| 45° | 1 | 1 | yes |
| 53° | 1 | 1 | yes |
| 60° | 1 | 1 | yes |
| 67.5° | 1 | 1 | yes |
| 75° | 1 | 1 | yes |
| 90° | 1 | 1 | yes |
| 120° | 1 | 1 | yes |
| 135° | 1 | 1 | yes |
| 150° | 1 | 1 | yes |
| 180° | 1 | 1 | yes |
| 210° | 1 | 1 | yes |
| 240° | 1 | 1 | yes |
| 270° | 1 | 1 | yes |
| 300° | 1 | 1 | yes |
| 330° | 1 | 1 | yes |
| 360° | 1 | 1 | yes |
The mistake students make
Writing sin²θ = 1 − cos θ. The cosine is squared as well: 1 − cos²θ. A second classic is reading sin²θ as sin(θ²) — it means (sin θ)².
Try it
- Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
- Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
- Cheat sheet entry for pythagorean identities.
Related identities
- 1 + tan²θ = sec²θPythagorean identity (tangent form)detailsproof
- 1 + cot²θ = csc²θPythagorean identity (cotangent form)detailsproof
- csc θ = 1/sin θReciprocal identity (cosecant)detailsproof
- sin²x = (1 − cos 2x)/2Power-reducing formula for sin²detailsproof
- cos²x = (1 + cos 2x)/2Power-reducing formula for cos²detailsproof
Category hub: Pythagorean identities · all identities: /identities