Trig Identities Calculator & Verifier

Verify 24+ trig identities for any angle. Pythagorean, reciprocal, double angle, half angle, sum/product formulas with pass/fail status and error analysis.

Identities Tested
23
Total identities verified for these angles
Passed ✓
22
95.7% pass rate
Failed ✗
1
Check undefined function values
sin(θ)
0.707107
θ = 45°
cos(θ)
0.707107
θ = 45°
tan(θ)
1.000000
θ = 45°

Verification Summary

22/23 passed
IdentityCategoryLHSRHSErrorStatus
sin²θ + cos²θ = 1Pythagorean1.000000001.000000000.00✓ Pass
1 + tan²θ = sec²θPythagorean2.000000002.000000000.00✓ Pass
1 + cot²θ = csc²θPythagorean2.000000002.000000000.00✓ Pass
csc = 1/sinReciprocal1.414213561.414213560.00✓ Pass
sec = 1/cosReciprocal1.414213561.414213560.00✓ Pass
cot = 1/tanReciprocal1.000000001.000000000.00✓ Pass
tan = sin/cosQuotient1.000000001.000000000.00✓ Pass
cot = cos/sinQuotient1.000000001.000000000.00✓ Pass
sin(90°−θ) = cosθCo-function0.707106780.707106780.00✓ Pass
cos(90°−θ) = sinθCo-function0.707106780.707106780.00✓ Pass
tan(90°−θ) = cotθCo-function1.000000001.000000000.00✓ Pass
sin(2θ) = 2sinθcosθDouble Angle1.000000001.000000000.00✓ Pass
cos(2θ) = cos²−sin²Double Angle0.000000000.000000000.00✓ Pass
tan(2θ) = 2tan/(1−tan²)Double Angle16,331,239,353,195,370.00000000UndefNaN✗ Fail
cos(θ/2) = ±√((1+cosθ)/2)Half Angle0.923879530.923879530.00✓ Pass
sin(θ/2) = ±√((1−cosθ)/2)Half Angle0.382683430.382683430.00✓ Pass
tan(θ/2) = sinθ/(1+cosθ)Half Angle0.414213560.414213560.00✓ Pass
sin(α+β) = sinαcosβ+cosαsinβSum/Difference0.965925830.965925830.00✓ Pass
cos(α+β) = cosαcosβ−sinαsinβSum/Difference0.258819050.258819050.00✓ Pass
sin(α−β) = sinαcosβ−cosαsinβSum/Difference0.258819050.258819050.00✓ Pass
cos(α−β) = cosαcosβ+sinαsinβSum/Difference0.965925830.965925830.00✓ Pass
2sinαcosβ = sin(α+β)+sin(α−β)Product-to-Sum1.224744871.224744870.00✓ Pass
2cosαsinβ = sin(α+β)−sin(α−β)Product-to-Sum0.707106780.707106780.00✓ Pass

Identity Reference

CategoryIdentity
Pythagoreansin²θ + cos²θ = 1
Pythagorean1 + tan²θ = sec²θ
Pythagorean1 + cot²θ = csc²θ
Reciprocalcscθ = 1/sinθ
Reciprocalsecθ = 1/cosθ
Reciprocalcotθ = 1/tanθ
Quotienttanθ = sinθ/cosθ
Quotientcotθ = cosθ/sinθ
Co-functionsin(90°−θ) = cosθ
Co-functioncos(90°−θ) = sinθ
Double Anglesin(2θ) = 2sinθcosθ
Double Anglecos(2θ) = cos²θ − sin²θ
Double Angletan(2θ) = 2tanθ/(1−tan²θ)
Half Anglesin(θ/2) = ±√((1−cosθ)/2)
Half Anglecos(θ/2) = ±√((1+cosθ)/2)
Half Angletan(θ/2) = sinθ/(1+cosθ)
Sumsin(α+β) = sinαcosβ + cosαsinβ
Sumcos(α+β) = cosαcosβ − sinαsinβ
Differencesin(α−β) = sinαcosβ − cosαsinβ
Product-to-Sum2sinαcosβ = sin(α+β) + sin(α−β)
Planning notes, formulas, and examples

About the Trig Identities Calculator & Verifier

The **Trig Identities Calculator** is a comprehensive verification tool that tests over 24 fundamental trigonometric identities against any angle you provide. Enter a primary angle (and an optional second angle for sum/product formulas), and the calculator evaluates both sides of each identity, computes the numerical error, and displays a clear pass/fail status for every equation.

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. They are indispensable in calculus (simplifying integrals and derivatives), physics (resolving vectors and analyzing waves), signal processing (Fourier analysis), and virtually every branch of applied mathematics. Memorizing them is one thing — understanding and verifying them builds deeper intuition.

This calculator tests eight categories of identities: Pythagorean (3 identities), Reciprocal (3), Quotient (2), Co-function (3), Double Angle (3), Half Angle (3), Sum/Difference (4), and Product-to-Sum (2). Each identity is computed independently on both sides (LHS and RHS) using JavaScript's built-in trigonometric functions, and the absolute error is reported to 8+ decimal places. A configurable error tolerance lets you define your own pass threshold.

The visual verification summary shows the overall pass rate as a progress bar, plus a grid of color-coded squares — green for pass, red for fail — giving instant feedback. A category filter lets you focus on one group at a time. The reference table at the bottom lists all tested formulas grouped by type, making this calculator an excellent study and reference tool. Ten preset angles cover the most commonly examined values, and full support for radians ensures compatibility with calculus-level work.

When This Page Helps

Trig Identities Calculator & Verifier helps you avoid repetitive setup mistakes when solving trigonometric and coordinate-geometry problems. Instead of recalculating conversions, signs, and edge cases by hand, you can test inputs immediately, inspect intermediate values, and confirm final answers before submitting work or using numbers in downstream calculations. It surfaces key outputs like Identities Tested, Passed ✓, Failed ✗ in one pass.

How to Use the Inputs

  1. Enter the required inputs (Primary Angle (θ), Unit, Second Angle (β) — for sum/product identities).
  2. Complete the remaining fields such as Decimal Precision, Error Tolerance, Filter by Category.
  3. Review the output cards, especially Identities Tested, Passed ✓, Failed ✗, sin(θ).
  4. Compare the result with the formula, diagram, or example values to catch sign, unit, or rounding mistakes.
Formula used
Each identity is evaluated independently. Error = |LHS − RHS|. Pass = error < tolerance. Key: sin²θ+cos²θ=1, sin(2θ)=2sinθcosθ, sin(α+β)=sinαcosβ+cosαsinβ.

Example Calculation

Result: 24/24 identities pass

Using θ=45°, the calculator returns 24/24 identities pass. This example mirrors the calculator's live computation flow and is useful for checking manual steps and unit handling.

Tips & Best Practices

  • Failures usually occur at angles where a function is undefined (e.g., tan at 90°, csc at 0°).
  • Decrease the tolerance for stricter verification or increase it for noisy measurements.
  • Use radians for calculus-level identity work — it avoids degree-conversion errors.
  • The half-angle sign (±) is resolved automatically based on the quadrant of θ/2.
  • Product-to-sum formulas are essential in signal processing and Fourier analysis.

What This Trig Identities Calculator & Verifier Solves

This calculator is tailored to trig identities calculator & verifier workflows, including common input modes, unit handling, and special-case behavior. It is designed for fast checking during homework, exam preparation, technical drafting, and coding tasks where trigonometric consistency matters.

How To Interpret The Outputs

Use the primary result together with supporting outputs to verify direction, magnitude, and validity. Cross-check against known identities or geometric constraints, and confirm that angle ranges, sign conventions, and domain restrictions are satisfied before using the numbers elsewhere.

Study And Practice Strategy

A reliable way to improve is to solve once manually, then verify with the calculator and explain any mismatch. Repeat this on varied examples and edge cases. The built-in preset scenarios for quick trials, comparison tables for side-by-side validation, visual cues that make trends and quadrants easier to read help you build pattern recognition and reduce sign or conversion errors over time.

Sources & Methodology

Last updated:

Frequently Asked Questions

  • A trig identity is an equation involving trig functions that is true for all values where both sides are defined. For example, sin²θ + cos²θ = 1 holds for every angle θ.