Trigonometric Identity Calculator

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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: sinยฒฮธ + cosยฒฮธ = 1 (true for ALL ฮธ)
Equation: sin ฮธ = 1/2 (true only for ฮธ = 30ยฐ, 150ยฐ, โ€ฆ)

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 โ€” The Foundation

The Pythagorean identities derive directly from the Pythagorean theorem applied to the unit circle. Since any point on the unit circle satisfies xยฒ + yยฒ = 1, with x = cosฮธ and y = sinฮธ:

sinยฒฮธ + cosยฒฮธ = 1
1 + tanยฒฮธ = secยฒฮธ   |   1 + cotยฒฮธ = cscยฒฮธ

Divide sinยฒฮธ + cosยฒฮธ = 1 by cosยฒฮธ for the tangent-secant form, by sinยฒฮธ 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, Aโˆ’B, 2A) in terms of functions of individual angles โ€” critical for calculus integration, Fourier analysis, and solving complex trig equations.

Sum & Difference

IdentityFormula
sin(A ยฑ B)sin A cos B ยฑ cos A sin B
cos(A ยฑ B)cos A cos B โˆ“ sin A sin B
tan(A ยฑ B)(tan A ยฑ tan B) / (1 โˆ“ tan A tan B)

Double Angle (set B = A above)

IdentityFormula
sin(2A)2 sin A cos A
cos(2A)cosยฒA โˆ’ sinยฒA = 2cosยฒA โˆ’ 1 = 1 โˆ’ 2sinยฒA
tan(2A)2 tan A / (1 โˆ’ tanยฒA)

How to Prove Trig Identities โ€” Strategy

  1. Work on ONE side only โ€” pick the more complex side. Never move terms across the equals sign.
  2. Convert to sin and cos โ€” replace tan, cot, sec, csc with their definitions.
  3. Apply Pythagorean identities โ€” replace sinยฒ + cosยฒ with 1, or 1 + tanยฒ with secยฒ.
  4. Factor and combine fractions over a common denominator.
  5. Use double angle / sum formulas when you see 2A, A+B, or Aโˆ’B patterns.
  6. Verify with a value โ€” plug in a specific angle to check both sides agree.
Prove tanยฒx โˆ’ sinยฒx = tanยฒx ยท sinยฒx
1. LHS = sinยฒx/cosยฒx โˆ’ sinยฒx  โ† convert tan to sin/cos
2. = sinยฒx ยท (1/cosยฒx โˆ’ 1)
3. = sinยฒx ยท (1 โˆ’ cosยฒx)/cosยฒx  โ† common denominator
4. = sinยฒx ยท sinยฒx/cosยฒx  โ† Pythagorean: 1 โˆ’ cosยฒx = sinยฒx
5. = sinยฒx ยท tanยฒx = RHS  โœ“

Complete Identity Quick Reference

CategoryKey FormulaCommon Use
Pythagoreansinยฒฮธ + cosยฒฮธ = 1Simplification, substitution
Double Anglesin(2ฮธ) = 2 sinฮธ cosฮธExpanding, integration
Half Anglesin(ฮธ/2) = ยฑโˆš((1โˆ’cosฮธ)/2)Exact values, integration
Sum to ProductsinA + sinB = 2sin((A+B)/2)cos((Aโˆ’B)/2)Factoring, signal processing
Product to SumsinA cosB = ยฝ[sin(A+B) + sin(Aโˆ’B)]Integration, Fourier
Negative Anglesin(โˆ’ฮธ) = โˆ’sinฮธSymmetry, simplification
Cofunctionsin(ฯ€/2 โˆ’ ฮธ) = cosฮธComplementary angles

Frequently asked

Yes. Click ๐Ÿ“ท Scan, upload a photo or PDF of your homework, and our AI extracts every trig identity along with both sides. Pick one to fill the form and prove.
Eight complete categories: Pythagorean, sum & difference, double angle, half angle, negative angle, sum-to-product, product-to-sum, and cofunction identities โ€” each with rendered LaTeX formulas.
Enter the LHS and RHS of the identity. Our AI verifies the identity first, then transforms one side step-by-step using known identities until it matches the other side. If false, it provides a counterexample.
Yes. The prover simplifies both sides independently first. If they differ, it reports โ€œNot a valid identityโ€ and shows a specific counterexample angle with numeric values for each side.
sinยฒฮธ + cosยฒฮธ = 1, derived from the Pythagorean theorem on the unit circle. Two related forms: 1 + tanยฒฮธ = secยฒฮธ and 1 + cotยฒฮธ = cscยฒฮธ.
They express trig functions of 2ฮธ in terms of ฮธ. sin(2A)=2sinAcosA; cos(2A) has three forms; tan(2A)=2tanA/(1โˆ’tanยฒA). Used for simplification, equation solving, and integration.
Work on one side only (the more complex one). Convert to sin and cos. Apply Pythagorean identities. Factor or combine fractions. Use double-angle or sum-to-product formulas as needed. Never cross the equals sign.