Fundamental Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable for which both sides are defined. Unlike equations that hold for only specific values, identities hold universally — they are tools, not puzzles to solve. The fundamental identities form the bedrock of all trigonometric manipulation: they let you rewrite expressions in simpler or more useful forms, prove other identities, and solve equations that would otherwise be intractable. The fundamental identities fall into several families: the reciprocal identities (defining cosecant, secant, and cotangent in terms of sine and cosine), the quotient identity (tangent as a ratio), and the Pythagorean identities (rooted in the unit-circle equation x² + y² = 1). Beyond these, the compound-angle (addition and subtraction) identities express the sine or cosine of a sum or difference of two angles in terms of the individual angles. Every double-angle, half-angle, and product-to-sum formula is derived directly from these compound-angle identities. Mastering fundamental identities is essential not only for high-school mathematics but also for calculus, physics, and engineering. In calculus, Pythagorean identities convert difficult integrals into standard forms. In physics, compound-angle identities describe wave interference. In this article you will see exactly where each identity comes from, how to apply them in algebraic simplification, and how they connect to the geometry of the unit circle.

Key takeaways
  • Reciprocal and Quotient Identities
  • The Pythagorean Identities
  • Even and Odd (Symmetry) Identities
  • Compound-Angle (Addition) Identities
  • Proving and Using Identities

Reciprocal and Quotient Identities

By definition on the unit circle, if P = (cos θ, sin θ), then the six trigonometric functions are: sin θ, cos θ, tan θ = sin θ / cos θ, csc θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = cos θ / sin θ. These six relationships are always true wherever the denominators are non-zero, and they are the starting point for every simplification. For example, 'sin θ · csc θ = 1' is an immediate consequence and is useful for cancellation.

The Pythagorean Identities

The unit circle has equation x² + y² = 1. Substituting x = cos θ and y = sin θ gives the first Pythagorean identity: sin²θ + cos²θ = 1. Dividing every term by cos²θ (where cos θ ≠ 0) yields tan²θ + 1 = sec²θ. Dividing the original by sin²θ (where sin θ ≠ 0) yields 1 + cot²θ = csc²θ. These three identities are the most frequently used in simplification and integration. Remember that sin²θ means (sin θ)², not sin(θ²).

Even and Odd (Symmetry) Identities

Cosine is an even function: cos(−θ) = cos θ. Sine is an odd function: sin(−θ) = −sin θ. Tangent is odd: tan(−θ) = −tan θ. These follow directly from the unit-circle symmetry about the x-axis. They are essential when simplifying expressions with negative angles or when working with inverse functions.

Compound-Angle (Addition) Identities

The compound-angle identities are: sin(A + B) = sin A cos B + cos A sin B; sin(A − B) = sin A cos B − cos A sin B; cos(A + B) = cos A cos B − sin A sin B; cos(A − B) = cos A cos B + sin A sin B. Notice the sign pattern: for sine, the sign on the right matches the sign of the compound angle; for cosine, the sign is reversed. A geometric proof uses the distance formula on the unit circle, but an elegant algebraic proof uses rotation matrices. These identities unlock double-angle and half-angle formulas: setting B = A in sin(A + B) immediately gives sin 2A = 2 sin A cos A, and cos 2A = cos²A − sin²A.

Proving and Using Identities

To prove an identity, work on one side only (usually the more complicated side) using known identities until you reach the other side. Never move terms across the equals sign — that would assume what you are trying to prove. Useful strategies: convert everything to sin and cos; factor expressions; multiply numerator and denominator by a conjugate; apply a Pythagorean identity to replace 1 − sin²θ with cos²θ or vice versa. When simplifying an expression (not proving), you may work on both sides independently.

The Key Equation

sin⁡2θ+cos⁡2θ=1tan⁡2θ+1=sec⁡2θ1+cot⁡2θ=csc⁡2θ\sin^2\theta + \cos^2\theta = 1 \qquad \tan^2\theta + 1 = \sec^2\theta \qquad 1 + \cot^2\theta = \csc^2\theta
  • \sin\theta — Sine (dimensionless): The y-coordinate of the point on the unit circle at angle θ.
  • \cos\theta — Cosine (dimensionless): The x-coordinate of the point on the unit circle at angle θ.
  • \tan\theta — Tangent (dimensionless): The ratio sin θ / cos θ; the slope of the radius at angle θ.
  • \sec\theta — Secant (dimensionless): The reciprocal of cosine: 1/cos θ. Undefined when cos θ = 0.
  • \csc\theta — Cosecant (dimensionless): The reciprocal of sine: 1/sin θ. Undefined when sin θ = 0.
  • \cot\theta — Cotangent (dimensionless): The ratio cos θ / sin θ; the reciprocal of tangent.

Worked Example

Problem. Prove the identity: (tan θ + cot θ) · sin θ · cos θ = 1

(tan⁡θ+cot⁡θ)sin⁡θcos⁡θ=sin⁡2θ+cos⁡2θsin⁡θcos⁡θ⋅sin⁡θcos⁡θ=1(\tan\theta + \cot\theta)\sin\theta\cos\theta = \frac{\sin^2\theta + \cos^2\theta}{\sin\theta\cos\theta}\cdot\sin\theta\cos\theta = 1

Answer. The identity is verified: both sides equal 1.

Figure
Diagram
Unit Circle and Pythagorean Identityxycos θsin θ1θP=(cos θ, sin θ)x² + y²= 1 ⟹ cos²θ + sin²θ = 1The radius = 1, so the hypotenuse is always 1.-111-1
Origins · 1 minAbū al-Wafāʾ Būzjānī and how this was discovered
AA
Abū al-Wafāʾ Būzjānī
940–998 · Persian

Abū al-Wafāʾ was one of the greatest mathematicians of the Islamic Golden Age. He introduced the tangent function into trigonometry, calculated the first tables of sine and cosine with remarkable precision (to eight decimal places), and was among the first to state the compound-angle identity for sine in a form that foreshadows the modern formula sin(A + B) = sin A cos B + cos A sin B. His work Kitāb al-Kāmil and his astronomical text Almagest commentary transmitted trigonometric knowledge that directly influenced European mathematicians. Abū al-Wafāʾ computed sine tables so accurately that they were not surpassed in precision for several centuries, all without a calculator — using only geometric constructions and iterative approximations.

  • The Unit Circle — Pythagorean and symmetry identities derive directly from the unit-circle definition of sine and cosine.
  • Double-Angle Identities — Setting A = B in the compound-angle formulas immediately produces sin 2A and cos 2A, making double-angle identities a direct consequence of the fundamental identities studied here.
  • Half-Angle Identities — Half-angle formulas are derived by rearranging the double-angle form of cos 2A, so they depend on the same Pythagorean and compound-angle identities.
  • Graphs of Trigonometric Functions — Even and odd identities explain the symmetry visible in the graphs of cosine (even, symmetric about y-axis) and sine (odd, symmetric about the origin).
  • Algebraic Proof Techniques — Proving trigonometric identities requires the same logical discipline as any algebraic proof: working from one side only and applying valid equivalences step by step.