Trig Identities

The Pythagorean identities

Three identities, all derived from the unit circle:

sin²θ + cos²θ = 1            (the fundamental)
1 + tan²θ = sec²θ              (divide by cos²θ)
cot²θ + 1 = csc²θ              (divide by sin²θ)

The first follows from the equation x² + y² = 1 of the unit circle. The other two come from dividing the first by cos²θ or sin²θ respectively. So three "identities" are really one identity in three forms.

Angle-sum and difference formulas

Function	Formula
sin(α + β)	sin α cos β + cos α sin β
sin(α − β)	sin α cos β − cos α sin β
cos(α + β)	cos α cos β − sin α sin β
cos(α − β)	cos α cos β + sin α sin β
tan(α + β)	(tan α + tan β) / (1 − tan α tan β)
tan(α − β)	(tan α − tan β) / (1 + tan α tan β)

These are the most-used identities after Pythagoras. Everything else (double-angle, half-angle, product-to-sum) follows from these. Memorize sin(α+β) and cos(α+β); the difference forms come from replacing β with −β; the tangent form comes from dividing the sin formula by the cos formula.

Double-angle formulas

Set α = β in the angle-sum formulas:

sin(2θ) = 2 sin θ cos θ
cos(2θ) = cos²θ − sin²θ
        = 2 cos²θ − 1
        = 1 − 2 sin²θ
tan(2θ) = 2 tan θ / (1 − tan²θ)

The three forms of cos(2θ) are equivalent (use sin² + cos² = 1 to convert between them); pick whichever simplifies your problem.

Half-angle formulas

Solving the double-angle cos(2θ) = 1 − 2 sin²θ for sin²θ — substitute θ = α/2:

sin²(α/2) = (1 − cos α) / 2
cos²(α/2) = (1 + cos α) / 2
tan(α/2)  = sin α / (1 + cos α) = (1 − cos α) / sin α

The square-root forms have ± signs depending on the quadrant of α/2 — be careful with signs. The tangent half-angle formula is always single-valued and is often the more useful form.

Sum-to-product and product-to-sum

Sum-to-product
sin α + sin β	= 2 sin((α+β)/2) cos((α−β)/2)
sin α − sin β	= 2 cos((α+β)/2) sin((α−β)/2)
cos α + cos β	= 2 cos((α+β)/2) cos((α−β)/2)
cos α − cos β	= −2 sin((α+β)/2) sin((α−β)/2)

Product-to-sum
2 sin α cos β	= sin(α + β) + sin(α − β)
2 cos α cos β	= cos(α + β) + cos(α − β)
−2 sin α sin β	= cos(α + β) − cos(α − β)

These are inverses — sum-to-product converts addition into multiplication; product-to-sum does the reverse. Used heavily in calculus (integrating sin·cos products), audio engineering (interference patterns), and physics (wave superposition).

Deriving identities with Euler's formula

Most trig identities have elegant complex-number derivations using e^(iθ) = cos θ + i sin θ.

Angle sum — multiply two complex exponentials:

e^(i(α+β)) = e^(iα) · e^(iβ)
cos(α+β) + i sin(α+β) = (cos α + i sin α)(cos β + i sin β)
                     = (cos α cos β − sin α sin β) + i(sin α cos β + cos α sin β)

Equate real and imaginary parts — both angle-sum formulas in one calculation. Without complex numbers, the same proofs require detailed geometric arguments.

Worked examples

Example 1 — exact value of sin(75°)

75° = 45° + 30°. Use angle-sum:

sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°
              = (√2/2)(√3/2) + (√2/2)(1/2)
              = (√6 + √2) / 4
              ≈ 0.966

Example 2 — simplifying for integration

Compute ∫sin²x dx. Direct integration is hard; use the half-angle identity:

sin²x = (1 − cos 2x) / 2
∫sin²x dx = ∫(1 − cos 2x)/2 dx = x/2 − sin(2x)/4 + C

Without the identity, this integral seems intractable. With it, one substitution.

Example 3 — beat frequencies in audio

Two close frequencies — sin(2π·440·t) + sin(2π·442·t). Use sum-to-product:

2 sin(2π·441·t) cos(2π·1·t)

The audible "beat" at 1 Hz (442 − 441 = 1, but for the difference we use (442−440)/2 = 1) is the cos factor — your ear hears 441 Hz volume modulating at 2 Hz. Used in piano tuning and audio engineering.

JavaScript: implementing identities

// Compute sin(2x) using identity (faster than Math.sin(2*x) sometimes)
function sin2x(x) {
  const s = Math.sin(x);
  const c = Math.cos(x);
  return 2 * s * c;
}

// cos(2x) — three equivalent forms
function cos2x(x) {
  const c = Math.cos(x);
  return 2 * c * c - 1;
  // or 1 - 2 * Math.sin(x)**2
  // or Math.cos(x)**2 - Math.sin(x)**2
}

// Verify identities (within float precision)
const angle = Math.PI / 7;
console.log(Math.sin(angle)**2 + Math.cos(angle)**2);  // ~1.0
console.log(Math.sin(2*angle), 2 * Math.sin(angle) * Math.cos(angle));  // equal

// Computing without trig calls — useful for performance-critical code
class TrigCache {
  constructor(angle) {
    this.s = Math.sin(angle);
    this.c = Math.cos(angle);
    this.s2x = 2 * this.s * this.c;
    this.c2x = this.c * this.c - this.s * this.s;
  }
}

Where trig identities show up

• Calculus integration. Powers of sin and cos integrate via half-angle formulas. Products integrate via product-to-sum. Without identities, these are intractable; with them, two-line solutions.
• Physics — wave superposition. Interference patterns, standing waves, beats — all use sum-to-product to combine sinusoids.
• Engineering — Fourier analysis. Signals decompose into sums of sinusoids; convolutions and frequency-domain operations use trig identities pervasively.
• Geometry. Proving "the sum of angles in a triangle is 180°," law of cosines, and other classical geometric results all use trig identities.
• Computer graphics. Rotation matrices, quaternion math for 3D rotation, cubic Bézier curves — all involve trig identity manipulations.
• Programming optimizations. Computing sin(x + Δ) by recurrence using sin(x), cos(x), and angle-sum formulas avoids repeated expensive trig calls in tight loops.

Common mistakes

• Wrong sign in difference formulas. sin(α − β) has − between the products; cos(α − β) has + between them. They're not symmetric. Easy to swap; learn by remembering sin's signs match the operation, cos's are opposite.
• Forgetting sign in half-angle. sin(α/2) = ±√(...) — the sign depends on which quadrant α/2 is in. Just taking the positive square root produces wrong signs in quadrants III and IV.
• Confusing sin²θ with sin(θ²). sin²θ means (sin θ)². sin(θ²) means sin of the angle θ². Very different functions. The notation is a historical artifact; doesn't follow standard function notation.
• Using identities outside their domain. tan(α + β) breaks down when 1 − tan α tan β = 0 (the formula has 0 in the denominator). At those angles you need to use limits or a different form.
• Forgetting the unit circle perspective. Memorizing identities without understanding leads to errors. Visualizing sin and cos as coordinates on the unit circle makes signs and symmetries obvious.
• Mixing radians and degrees in formulas. All these identities work in both, but you can't mix — sin(α°+ β rad) is meaningless. Always convert to one unit before applying identities.