Precalculus Advanced

Trig Identities Verifier

Numerically verify any trigonometric identity by evaluating both sides at 10 random test points — and explore a complete reference library of Pythagorean, reciprocal, sum/difference, double-angle, half-angle, and more.

Live Verifier · Identity Library · Precalculus
Trig Identity Tool
Left-Hand Side (LHS)
=
Right-Hand Side (RHS)
Use: sin(x), cos(x), tan(x), x^2, sqrt(x), PI, * for multiply.
csc = 1/sin(x)   sec = 1/cos(x)   cot = cos(x)/sin(x)
Quick Examples

Click any identity card to auto-load it into the Identity Verifier (Mode 1) for numerical verification.

Pythagorean Identities ▼
sin²θ + cos²θ = 1
Fundamental Pythagorean identity
Click to verify →
1 + tan²θ = sec²θ
Divide sin²+cos²=1 by cos²
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1 + cot²θ = csc²θ
Divide sin²+cos²=1 by sin²
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sin²θ = 1 − cos²θ
Rearrangement of Pythagorean
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Reciprocal Identities ▼
csc θ = 1 / sin θ
Cosecant is reciprocal of sine
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sec θ = 1 / cos θ
Secant is reciprocal of cosine
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cot θ = 1 / tan θ
Cotangent is reciprocal of tangent
Click to verify →
sin θ · csc θ = 1
Product of reciprocals equals 1
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Quotient Identities ▼
tan θ = sin θ / cos θ
Tangent defined as ratio
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cot θ = cos θ / sin θ
Cotangent defined as ratio
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Co-function Identities ▼
sin(π/2 − θ) = cos θ
Sine and cosine are co-functions
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cos(π/2 − θ) = sin θ
Cosine and sine are co-functions
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tan(π/2 − θ) = cot θ
Tangent and cotangent are co-functions
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sin²(π/2−θ) + sin²θ = 1
Co-function + Pythagorean combined
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Even / Odd Identities ▼
cos(−θ) = cos θ
Cosine is an even function
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sin(−θ) = −sin θ
Sine is an odd function
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tan(−θ) = −tan θ
Tangent is an odd function
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cos²(−θ) + sin²(−θ) = 1
Pythagorean holds for negative angles
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Sum & Difference Formulas ▼
sin(a+b) = sin a cos b + cos a sin b
Sine addition formula — uses two variables x, y
Click to verify (uses x and y) →
sin(a−b) = sin a cos b − cos a sin b
Sine subtraction formula
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cos(a+b) = cos a cos b − sin a sin b
Cosine addition formula
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cos(a−b) = cos a cos b + sin a sin b
Cosine subtraction formula
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tan(a+b) = (tan a + tan b) / (1 − tan a tan b)
Tangent addition formula
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tan(a−b) = (tan a − tan b) / (1 + tan a tan b)
Tangent subtraction formula
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Double-Angle Formulas ▼
sin(2θ) = 2 sin θ cos θ
Sine double angle
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cos(2θ) = cos²θ − sin²θ
Cosine double angle — form 1
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cos(2θ) = 2cos²θ − 1
Cosine double angle — form 2
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cos(2θ) = 1 − 2sin²θ
Cosine double angle — form 3
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tan(2θ) = 2tanθ / (1 − tan²θ)
Tangent double angle
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sin²θ = (1 − cos 2θ) / 2
Power-reduction formula for sin²
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cos²θ = (1 + cos 2θ) / 2
Power-reduction formula for cos²
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Half-Angle Formulas ▼
sin²(θ/2) = (1 − cos θ) / 2
Half-angle for sin² (always valid)
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cos²(θ/2) = (1 + cos θ) / 2
Half-angle for cos² (always valid)
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sin(θ/2) = ±√((1−cosθ)/2)
Half-angle for sin (sign depends on quadrant)
Click to verify (x in [0,2π]) →
cos(θ/2) = ±√((1+cosθ)/2)
Half-angle for cos (sign depends on quadrant)
Click to verify (x in [0,2π]) →
tan(θ/2) = sin θ / (1 + cos θ)
Half-angle tangent — form 1
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tan(θ/2) = (1 − cos θ) / sin θ
Half-angle tangent — form 2
Click to verify →
Product-to-Sum Formulas ▼
sin a cos b = ½[sin(a+b) + sin(a−b)]
Product-to-sum — sin·cos
Click to verify (x, y) →
cos a cos b = ½[cos(a−b) + cos(a+b)]
Product-to-sum — cos·cos
Click to verify (x, y) →
sin a sin b = ½[cos(a−b) − cos(a+b)]
Product-to-sum — sin·sin
Click to verify (x, y) →
Result
Enter LHS and RHS above, then press Verify Identity — or browse the Reference Library tab and click any identity to load it here.
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First Counterexample
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Expressions Tested
LHS: —
RHS: —
Test Values (10 random x in [0.1, 2π−0.1])
x (rad) LHS value RHS value |LHS−RHS| Match?
Verification Steps & Interpretation
Verifying vs. Proving an Identity
Numerical check ≠ Algebraic proof

This tool checks both sides of an identity at 10 random values of x. If they match within a tolerance of 10⁻⁸, it reports "Verified." But this is not a proof.

A numerical check can:

  • Confirm an identity is likely true — and build intuition.
  • Definitively disprove a false identity by finding a counterexample.

To prove an identity algebraically, start with one side and use known identities (Pythagorean, quotient, etc.) to transform it step-by-step until it matches the other side. You should never move terms across the equal sign.

Tip: Use this tool to check your answer before writing the algebraic proof. If it says "Not an Identity," you know to re-examine your work.
How to Use Trig Identities
Simplify · Verify · Substitute

Trig identities appear in three main settings:

  • Simplification — Replace a complex expression with a simpler one. E.g., replace sin²x + cos²x with 1 anywhere it appears.
  • Verification — Confirm that two expressions are equivalent by transforming one into the other without crossing the = sign.
  • Substitution — In integrals (calculus), substitute to convert difficult forms. E.g., replace √(1−sin²x) with |cos x|.
Strategy: always work on the more complex side first. Try converting everything to sin and cos, and look for Pythagorean or double-angle patterns.
Related Calculators
Sum & Difference Formulas
Evaluate sin(a±b), cos(a±b), tan(a±b) exactly using special angles.
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Double & Half-Angle Calculator
Find exact values using double-angle and half-angle formulas.
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Unit Circle Explorer
Visualize all standard angles and their exact trig values on the unit circle.
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Trig Equations Solver
Solve equations like 2sin(x)−1=0 and find all solutions in [0, 2π].
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