Pythagorean Identities: sin²+cos²=1 Full Proofs & Uses

The Pythagorean identities are three equations that tie the trigonometric functions together: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ. The first is the most used identity in all of trigonometry, and the other two follow from it by division.

They matter because they let you find one ratio from another without knowing the angle. Given sin θ = 3/5, the identity immediately reveals cos θ, which is the starting point for simplifying expressions, solving equations and integrating in calculus.

Key Concepts

• sin²θ means (sin θ)², the square of the sine value, not the sine of θ².
• Unit circle proof: the point (cos θ, sin θ) lies on the circle x² + y² = 1, so cos²θ + sin²θ = 1 directly.
• Right-triangle proof: with hypotenuse 1, the legs are sin θ and cos θ, and Pythagoras gives sin²θ + cos²θ = 1.
• Derived identities: dividing sin²θ + cos²θ = 1 by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ gives 1 + cot²θ = csc²θ.
• Domains: the second identity needs cos θ ≠ 0, the third needs sin θ ≠ 0.

Formulas

• Main identity: sin²θ + cos²θ = 1, valid for every real angle θ.
• Tangent form: 1 + tan²θ = sec²θ, valid wherever cos θ ≠ 0.
• Cotangent form: 1 + cot²θ = csc²θ, valid wherever sin θ ≠ 0.
• Rearrangements: sin²θ = 1 − cos²θ and cos²θ = 1 − sin²θ, used to eliminate one function from an expression.

Worked Example

If cos θ = 5/13 and θ is acute, find sin θ and tan θ.

1. Use sin²θ = 1 − cos²θ = 1 − 25/169 = 144/169.
2. Take the square root: sin θ = ±12/13. Since θ is acute (first quadrant), sine is positive, so sin θ = 12/13.
3. Recognize the 5-12-13 triangle as a sanity check: 5² + 12² = 25 + 144 = 169 = 13².
4. tan θ = sin θ / cos θ = (12/13) / (5/13) = 12/5.
5. Verify with the tangent identity: 1 + tan²θ = 1 + 144/25 = 169/25, and sec²θ = (13/5)² = 169/25. Both sides match.

Common Misconceptions

• sin²θ is not sin(θ²). The square applies to the function value, never to the angle inside.
• Square roots give two signs. sin θ = ±√(1 − cos²θ); the quadrant of θ decides which sign survives.
• The identities do not determine θ. They relate the functions to each other; the angle itself needs more information.

Real-World Applications

• Simplifying expressions: replacing sin²θ + cos²θ with 1 collapses long trigonometric expressions.
• Finding unknown ratios: completing the set of six functions from a single known value.
• Calculus: the substitution sin²θ = 1 − cos²θ is central to integrating powers of sine and cosine.
• Physics: checking that resolved components of a vector satisfy Fx² + Fy² = F².

The Pythagorean identities, led by sin²θ + cos²θ = 1, bind the trigonometric functions into one family. Prove it once from the unit circle, divide to get the other two forms, and always let the quadrant choose the sign after a square root.