Trigonometric Identity Calculator
sin(x)^2 + cos(x)^2
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What Are Trigonometric Identities?
Trigonometric identities are equations involving trig functions that hold true for all values of the variable where both sides are defined. Unlike trig equations (true only for specific angles), identities are universal truths. They are essential for simplifying expressions, solving equations, evaluating integrals, and proving new results in mathematics, physics, and engineering.
Identity vs Equation
Identity: sin²θ + cos²θ = 1 (true for ALL θ)
Equation: sin θ = 1/2 (true only for θ = 30°, 150°, …)
Why They Matter
Simplify complex expressions, solve trig equations, evaluate calculus integrals, model wave interference, and prove mathematical theorems.
Our 8 Categories
Pythagorean, Sum & Difference, Double Angle, Half Angle, Negative Angle, Sum-to-Product, Product-to-Sum, Cofunction.
The Pythagorean Identities — The Foundation
The Pythagorean identities derive directly from the Pythagorean theorem applied to the unit circle. Since any point on the unit circle satisfies x² + y² = 1, with x = cosθ and y = sinθ:
Divide sin²θ + cos²θ = 1 by cos²θ for the tangent-secant form, by sin²θ for the cotangent-cosecant form. These three are the most-used identities in all of trigonometry.
Double Angle & Sum/Difference Formulas
These let you express trig functions of combined angles (A+B, A−B, 2A) in terms of functions of individual angles — critical for calculus integration, Fourier analysis, and solving complex trig equations.
Sum & Difference
| Identity | Formula |
|---|---|
| sin(A ± B) | sin A cos B ± cos A sin B |
| cos(A ± B) | cos A cos B ∓ sin A sin B |
| tan(A ± B) | (tan A ± tan B) / (1 ∓ tan A tan B) |
Double Angle (set B = A above)
| Identity | Formula |
|---|---|
| sin(2A) | 2 sin A cos A |
| cos(2A) | cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A |
| tan(2A) | 2 tan A / (1 − tan²A) |
How to Prove Trig Identities — Strategy
- Work on ONE side only — pick the more complex side. Never move terms across the equals sign.
- Convert to sin and cos — replace tan, cot, sec, csc with their definitions.
- Apply Pythagorean identities — replace sin² + cos² with 1, or 1 + tan² with sec².
- Factor and combine fractions over a common denominator.
- Use double angle / sum formulas when you see 2A, A+B, or A−B patterns.
- Verify with a value — plug in a specific angle to check both sides agree.
1. LHS = sin²x/cos²x − sin²x ← convert tan to sin/cos
2. = sin²x · (1/cos²x − 1)
3. = sin²x · (1 − cos²x)/cos²x ← common denominator
4. = sin²x · sin²x/cos²x ← Pythagorean: 1 − cos²x = sin²x
5. = sin²x · tan²x = RHS ✓
Complete Identity Quick Reference
| Category | Key Formula | Common Use |
|---|---|---|
| Pythagorean | sin²θ + cos²θ = 1 | Simplification, substitution |
| Double Angle | sin(2θ) = 2 sinθ cosθ | Expanding, integration |
| Half Angle | sin(θ/2) = ±√((1−cosθ)/2) | Exact values, integration |
| Sum to Product | sinA + sinB = 2sin((A+B)/2)cos((A−B)/2) | Factoring, signal processing |
| Product to Sum | sinA cosB = ½[sin(A+B) + sin(A−B)] | Integration, Fourier |
| Negative Angle | sin(−θ) = −sinθ | Symmetry, simplification |
| Cofunction | sin(π/2 − θ) = cosθ | Complementary angles |