Trigonometric Identities
Essential trig identities: Pythagorean identities, reciprocal identities, and quotient identities.
A complete reference for simplifying trig expressions.
Pythagorean Identities
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
These are the most important trig identities.
They are all derived from the Pythagorean theorem applied to the unit circle.
Reciprocal Identities
| Function | Reciprocal |
|---|---|
| sin θ | 1 / csc θ |
| cos θ | 1 / sec θ |
| tan θ | 1 / cot θ |
| csc θ | 1 / sin θ |
| sec θ | 1 / cos θ |
| cot θ | 1 / tan θ |
Quotient Identities
- tan θ = sin θ / cos θ
- cot θ = cos θ / sin θ
Example 1
If sin θ = 3/5 and θ is in the first quadrant, find cos θ
Using: sin²θ + cos²θ = 1
(3/5)² + cos²θ = 1
9/25 + cos²θ = 1
cos²θ = 1 - 9/25 = 16/25
cos θ = 4/5 (positive because θ is in the first quadrant)
Example 2
Simplify: sin²θ × sec²θ + cos²θ × sec²θ
Factor out sec²θ: sec²θ × (sin²θ + cos²θ)
Since sin²θ + cos²θ = 1:
= sec²θ × 1
= sec²θ
When to Use It
Use trigonometric identities when:
- Simplifying complex trig expressions
- Proving that two trig expressions are equal
- Finding unknown trig values when one value is known
- Solving trig equations by rewriting them in simpler forms
Key Notes
- Three Pythagorean identities: sin²θ + cos²θ = 1 (fundamental); tan²θ + 1 = sec²θ (divide by cos²θ); 1 + cot²θ = csc²θ (divide by sin²θ). All three are derived from the unit circle definition.
- Quotient identities: tanθ = sinθ/cosθ and cotθ = cosθ/sinθ. These connect the four secondary trig functions directly to sine and cosine.
- Reciprocal identities: cscθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ. These are definitions, not results to be proved — they follow directly from the names of the functions.
- Even and odd functions: cos(−θ) = cosθ (even); sin(−θ) = −sinθ (odd); tan(−θ) = −tanθ (odd). These symmetry properties are useful when simplifying expressions with negative angles.
- Verifying vs solving: To verify an identity, work on one side only (or both sides independently) until they match. Never move terms across the equals sign — that assumes what you are trying to prove.