TRIGONOMETRYTrigonometryMathematics Calculator
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Trigonometric Identities

Verify sin²+cos²=1, 1+tan²=sec², 1+cot²=csc², sin/cos=tan, cos/sin=cot for any angle. All identities hold for every valid input.

Concept Fundamentals
sin²θ+cos²θ=1
Pythagorean
1+tan²θ=sec²θ
Tangent-Secant
1+cot²θ=csc²θ
Cot-Csc
tan=sin/cos, cot=cos/sin
Quotient

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sin²θ+cos²θ=1 always — the most fundamental identity. Use it to find one from the other. 1+tan²θ=sec²θ when cosθ≠0. 1+cot²θ=csc²θ when sinθ≠0. Reciprocal: csc=1/sin, sec=1/cos, cot=1/tan. Quotient: tan=sin/cos, cot=cos/sin.

Key quantities
sin²θ+cos²θ=1
Pythagorean
Key relation
1+tan²θ=sec²θ
Tangent-Secant
Key relation
1+cot²θ=csc²θ
Cot-Csc
Key relation
tan=sin/cos, cot=cos/sin
Quotient
Key relation

Ready to run the numbers?

Why: Identities are the backbone of trig simplification. sin²+cos²=1 is the Pythagorean theorem on the unit circle. The others follow from definitions.

How: Pythagorean: from x²+y²=1 on unit circle with (cos,sin). Divide by cos² for 1+tan²=sec², by sin² for 1+cot²=csc². Quotient: tan=sin/cos by definition.

sin²θ+cos²θ=1 always — the most fundamental identity. Use it to find one from the other.1+tan²θ=sec²θ when cosθ≠0. 1+cot²θ=csc²θ when sinθ≠0.
Sources:Wolfram MathWorld — Trigonometric IdentitiesKhan Academy — Trig Identities

Run the calculator when you are ready.

Start CalculatingEnter an angle to verify all fundamental trig identities

Examples — Click to Load

trig-identities.sh
CALCULATED
$ verify --angle 30° --all-identities
sin²+cos²
1 ✓
1+tan²=sec²
✓
1+cot²=csc²
✓
Overall
✓ All OK
sin(θ)
0.5
cos(θ)
0.8660254
tan(θ)
0.57735027
Quadrant
Q1
Share:
Trig Identities Verification
θ = 30°
sin²+cos² = 1 ✓1+tan²=sec² ✓1+cot²=csc² ✓
All identities verified ✓
numbervibe.com/calculators/mathematics/trigonometry/trig-identities-calculator

Trig Value Breakdown

All 6 Trig Functions

sin² vs cos² (Pythagorean Identity)

Calculation Breakdown

INPUT
Input Angle
30°
TRIG VALUES
sin(θ)
0.5
sin(30°)
cos(θ)
0.8660254
cos(30°)
tan(θ)
0.57735027
\text{sin}/\text{cos}
csc(θ)
2
1/\text{sin}
sec(θ)
1.15470054
1/\text{cos}
cot(θ)
1.73205081
\text{cos}/\text{sin}
PYTHAGOREAN IDENTITIES
sin²θ + cos²θ
1
0.25 + 0.75 = 1
Verification
✓ Verified
\text{sin}^{2}+\text{cos}^{2} = 1
PYTHAGOREAN IDENTITIES
1 + tan²θ
1.33333333
1 + 0.33333333
sec²θ
1.33333333
sec²(30°)
1+tan²=sec²
✓ Verified
PYTHAGOREAN IDENTITIES
1 + cot²θ
4
1 + cot²(30°)
csc²θ
4
1/\text{sin}^{2}\text{theta}
1+cot²=csc²
✓ Verified
QUOTIENT IDENTITIES
sin θ / cos θ
0.57735027
0.5 / 0.8660254
tan θ
0.57735027
\text{sin}/\text{cos} = \text{tan}
sin/cos=tan
✓ Verified
QUOTIENT IDENTITIES
cos θ / sin θ
1.73205081
0.8660254 / 0.5
cot θ
1.73205081
\text{cos}/\text{sin} = ext{cot}
cos/sin=cot
✓ Verified
RECIPROCAL IDENTITIES
csc θ = 1/sin θ
2 = 1/0.5
ext{Reciprocal} ext{identity}
sec θ = 1/cos θ
1.15470054 = 1/0.8660254
ext{Reciprocal} ext{identity}
cot θ = 1/tan θ
1.73205081 = 1/0.57735027
ext{Reciprocal} ext{identity}
COFUNCTION IDENTITIES
sin(90°−θ) = cos θ
0.8660254 = 0.8660254
ext{Cofunction}
cos(90°−θ) = sin θ
0.5 = 0.5
ext{Cofunction}
SUMMARY
Overall Status
✓ All identities verified

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

✓

sin²θ+cos²θ=1 is the Pythagorean identity — holds for every angle θ.

— Wolfram MathWorld

≡

All other trig identities derive from Pythagorean, quotient, and reciprocal definitions.

— Khan Academy

Key Takeaways

  • • sin²θ + cos²θ = 1 — The fundamental Pythagorean identity holds for every angle on the unit circle
  • • 1 + tan²θ = sec²θ — Derived by dividing sin²+cos²=1 by cos²θ; undefined when cos θ = 0 (90°, 270°)
  • • 1 + cot²θ = csc²θ — Derived by dividing sin²+cos²=1 by sin²θ; undefined when sin θ = 0 (0°, 180°)
  • • tan θ = sin θ / cos θ and cot θ = cos θ / sin θ — Quotient identities define tangent and cotangent
  • • All six trig functions (sin, cos, tan, csc, sec, cot) are interconnected through these identities

Did You Know?

📐The Pythagorean identity sin²+cos²=1 comes directly from the unit circle: x²+y²=1 where x=cos θ and y=sin θSource: Wolfram MathWorld
🔗All trig identities can be derived from just sin²+cos²=1 and the definitions of tan, csc, sec, cotSource: Khan Academy
🌍Trig identities are used in GPS, satellite navigation, and signal processing — verifying them numerically is crucialSource: MIT OpenCourseWare
🎯At 45°, sin²θ = cos²θ = 0.5 exactly, so the Pythagorean identity is perfectly balancedSource: Paul's Online Notes
⚡Computer graphics and game engines rely on trig identities for efficient rotation and transformation calculationsSource: IEEE Graphics
📊Fourier analysis uses trig identities to decompose signals into sine and cosine componentsSource: NIST DLMF

How Trig Identities Work

Trigonometric identities are equations that hold true for all valid angles. They allow you to simplify expressions, solve equations, and verify computations. This calculator numerically verifies the key identities for any angle you enter.

Pythagorean Identities

sin²θ + cos²θ = 1 is the foundation. Divide by cos²θ to get 1 + tan²θ = sec²θ, or by sin²θ to get 1 + cot²θ = csc²θ. These hold wherever the denominators are nonzero.

Quotient Identities

tan θ = sin θ / cos θ and cot θ = cos θ / sin θ define tangent and cotangent as ratios. At 90° and 270°, cos θ = 0 so tan is undefined. At 0° and 180°, sin θ = 0 so cot is undefined.

Reciprocal & Cofunction Identities

csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ. Cofunctions relate complementary angles: sin(90°−θ) = cos θ and cos(90°−θ) = sin θ.

Expert Tips

Verify Before Simplifying

When simplifying trig expressions, plug in a test angle to verify your result. Use this calculator to check that both sides match.

Watch for Undefined Cases

1+tan²=sec² fails at 90° and 270° (cos=0). 1+cot²=csc² fails at 0° and 180° (sin=0). The calculator shows N/A for these.

Floating-Point Tolerance

Computers use floating-point math. We consider an identity verified if the difference is < 10⁻¹⁰. Tiny rounding errors are expected.

Use With Unit Circle

Pair this with the Unit Circle Calculator to visualize why sin²+cos²=1.

Why Use This Calculator vs. Other Tools?

FeatureThis CalculatorManual VerificationSymbolic CAS
All 5 identity types verified✅❌ Time-consuming✅
Actual computed values shown✅✅⚠️ Symbolic only
sin²+cos², 1+tan²=sec², 1+cot²=csc²✅✅✅
Quotient identities sin/cos=tan✅✅✅
Handles undefined (90°, 0°)✅ N/A shown⚠️ Care needed✅
Visual charts & breakdown✅❌❌
Step-by-step with sections✅✅❌
Copy & share results✅❌❌

Frequently Asked Questions

Why does sin²+cos² always equal 1?

On the unit circle, the point at angle θ has coordinates (cos θ, sin θ). By the Pythagorean theorem, x² + y² = 1 for any point on the circle. So cos²θ + sin²θ = 1.

When is 1+tan²=sec² undefined?

When cos θ = 0, i.e., at 90° and 270°. At these angles, tan θ and sec θ are undefined (division by zero), so the identity does not apply.

When is 1+cot²=csc² undefined?

When sin θ = 0, i.e., at 0° and 180°. At these angles, cot θ and csc θ are undefined, so the identity does not apply.

What does "verified" mean?

We compute both sides numerically. If the difference is less than 10⁻¹⁰ (floating-point tolerance), we mark it as verified. This accounts for tiny rounding errors.

Can I use radians?

Yes. Select "Radians" and enter values like π/4 or 1.5708. All identities hold in radians as well.

Why do some identities show N/A?

At angles where a function is undefined (e.g., tan at 90°), we cannot verify that identity. N/A means "not applicable" for that angle.

How are quotient identities derived?

tan θ = sin θ / cos θ by definition (opposite/adjacent in a right triangle). cot θ = cos θ / sin θ = 1/tan θ. These are the quotient identities.

What are cofunction identities?

sin(90°−θ) = cos θ and cos(90°−θ) = sin θ. Complementary angles swap sine and cosine. Similarly, tan(90°−θ) = cot θ.

Trig Identities by the Numbers

5
Identity Types Verified
6
Trig Functions
1
Pythagorean Foundation
∞
Valid Angles

Disclaimer: This calculator uses IEEE 754 floating-point arithmetic. Verification is based on a tolerance of 10⁻¹⁰. Results are for educational purposes. For symbolic proofs, use a computer algebra system. Not a substitute for formal mathematical verification.

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