Pythagorean identities · proof ·what it is and when to use it
Proof: sin²θ + cos²θ = 1
Pythagorean identity — proved by unit circle. Every line below says which rule it uses, so nothing has to be taken on faith.
The proof
Take the point P = (cos θ, sin θ) on the unit circle and drop a perpendicular to the x-axis. The right triangle formed has legs |cos θ| and |sin θ| and hypotenuse equal to the radius, 1.
- leg₁ = cos θ, leg₂ = sin θ, hypotenuse = 1Definition of sine and cosine on the unit circle.
- cos²θ + sin²θ = 1²Pythagoras applied to that triangle.
- sin²θ + cos²θ = 1Same equation, reordered. Holds for every θ because the construction works in all four quadrants (squares absorb the signs).
The right-triangle picture only covers acute angles; the unit-circle version is what makes the identity true for every real angle, including 200° and −π/3.
Where the proof stops applying
Both sides are undefined at the same angles — where a denominator in the proof reaches zero (for instance cos x = 0 in a tangent or secant form). Elsewhere the argument above holds for every real angle.
The same claim, checked numerically
A proof is not the same thing as a check, and this page shows both: the reasoning above, and the first few angles fed to the same engine behind the verifier. If they ever disagreed, the data would be broken — the build would fail before publishing (accuracy policy).
| 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 |
Related
- How would I find sin²θ + cos²θ = 1 myself? — the derivation, which is a different question from the proof.
- Pythagorean identity: when to use it — the practical side.
- All pythagorean identities · proof index