Which double-angle identities does the explorer cover?

All six standard identities: sin(2 theta), cos(2 theta), tan(2 theta), csc(2 theta), sec(2 theta), and cot(2 theta). Each has its own tab and its own derivation view, either geometric or algebraic.

Why do only sin and cos get geometric proofs?

Sine and cosine are the foundational identities. Once you have them, tangent comes from sine over cosine, and cosecant, secant, and cotangent are reciprocals. The tool reflects this structure by limiting geometric proofs to sin and cos and treating the other four as algebraic consequences.

How do I follow the animated proof?

Use the toolbar below the diagram. Play steps through all six stages automatically. Prev and Next move one stage at a time. The speed selector switches between 0.5x, 1x, 1.5x, and 2x. The right panel describes each step in order.

Can I verify an identity at a specific angle?

Yes. Move the theta slider to the angle you want, then read the two metric cards near the bottom of the scene. They show the values of both sides of the identity, computed independently. The formula table at the bottom does the same for all six functions at once.

What does the source column of the formula table mean?

It tells you how each identity was obtained. Sine and cosine are labeled geometric because they are proved directly from a triangle. Tangent is labeled via sin and cos, while cosecant, secant, and cotangent are labeled via their reciprocal partners.

Double Angle Identities: Interactive Proofs

Key Terms

• Double-angle identity — a formula expressing a trig function of

2\theta

in terms of trig functions of

\theta

.
• Geometric proof — a derivation that uses a drawing (an isosceles triangle, two radii, a bisector) rather than algebra.
• Bisector — a line that splits an angle into two equal halves. In this tool,

OM

bisects the apex angle

2\theta

.
• Reciprocal identity — a function written as

1/\text{(another function)}

.

\csc = 1/\sin

,

\sec = 1/\cos

,

\cot = 1/\tan

.
• Derived identity — one obtained by combining or rearranging others.

\tan(2\theta)

,

\csc(2\theta)

,

\sec(2\theta)

,

\cot(2\theta)

all follow from the formulas for

\sin(2\theta)

and

\cos(2\theta)

.

Switching Between Functions

A row of six tabs at the top lets you select which double-angle identity to study:

\sin(2\theta)

,

\cos(2\theta)

,

\tan(2\theta)

,

\csc(2\theta)

,

\sec(2\theta)

,

\cot(2\theta)

.

How selection changes the view:
•

\sin

and

\cos

open the geometric proof scene with a step-by-step animation.
•

\tan

,

\csc

,

\sec

, and

\cot

open the derived identity card with the algebraic chain.
• The active tab is highlighted in deep blue.
• The URL updates with

?fn=...

, so links you copy preserve the selected function.

You can also click any row of the formula table at the bottom to jump directly to that function.

Adjusting the Angle θ

Each view exposes a slider for the base angle

\theta

in degrees, between

10°

and

80°

.

What changes as you slide:
• On geometric scenes, the SVG triangle resizes and reshapes in real time.
• The number readout next to the slider shows the exact degree value.
• The verification cards at the bottom recompute both sides of the identity using the new

\theta

.

Slow sweeps near

45°

are useful for seeing how the relationships behave in the most symmetric case, while values near the extremes (

10°

or

80°

) show how the same identities still hold for narrow and wide triangles.

Playing Through a Geometric Proof

When

\sin

or

\cos

is active, an animated proof unfolds in six steps. A toolbar gives you control:

• Reset — return to step 0 with a blank scene.
• Prev / Next — step through one stage at a time.
• Play / Pause — advance automatically.
• Speed selector —

0.5\times

,

1\times

,

1.5\times

, or

2\times

.

Each step adds one geometric element (radii, triangle fill, bisector, half-angles, leg labels, final metrics). The right-hand panel logs each step's name and rationale, so you can stop and re-read at any point.

Reading the Geometric Scene

The SVG shows the unit circle with two radii

OA

and

OB

of length

1

meeting at the center

O

with angle

2\theta

between them.

Elements that appear across the steps:
• Red arc at

O

— the apex angle, labeled

C = 2\theta

.
• Indigo chord

AB

— the base of the isosceles triangle.
• Blue segment

OM

— the perpendicular bisector, equal to

\cos\theta

.
• Half-angles at

O

— each labeled

\theta

once the bisector is drawn.
• Half-chord labels — each labeled

\sin\theta

on segments

MA

and

MB

.

A small right-angle mark appears at

M

when the perpendicular bisector becomes visible.

Working with Derived Identities

Selecting

\tan(2\theta)

,

\csc(2\theta)

,

\sec(2\theta)

, or

\cot(2\theta)

opens a different card layout. Instead of a triangle, the page shows the algebraic derivation as a chain of equations.

Layout of the derived card:
• A short intro explains which earlier identity the current one rests on.
• Jump buttons link directly to the geometric proofs of the source identities.
• A multi-line derivation block shows each manipulation with a brief side note.
• Verification cards confirm both sides match numerically.

This split keeps the geometric ideas isolated to two functions and treats the other four as quick algebraic consequences.

Reading the Formula Table

A reference table beneath every scene lists all six identities at once:

• Function column — the name of the trig function with

2\theta

argument.
• Identity column — the right-hand side of the formula.
• Value column — the numeric value computed at the current

\theta

.
• Source column — labels each identity as geometric (

\sin

,

\cos

) or via X for derived ones.

Click any row to make that function active. The current selection gets a deep-blue left border and a tinted background, making it easy to track context.

Verifying Identities Numerically

Every scene includes two metric cards near the bottom that compute both sides of the active identity at the current

\theta

.

Example for

\sin(2\theta)

:
• Left card shows

\sin(2\theta)

.
• Right card shows

2\sin\theta\cos\theta

.

The two numbers always match (within rounding to three decimals). Sweeping the slider while watching the cards is a fast empirical check that the identity holds for every angle, not just the one in the picture. The formula table mirrors this behavior across all six functions simultaneously.

Geometric Proofs: sin(2θ) and cos(2θ)

The two foundational identities are proved by drawing an isosceles triangle with two unit radii.

sin(2θ) = 2 sin θ cos θ — area is computed two ways:

\text{area} = \tfrac{1}{2}\sin(2\theta) = \sin\theta\cos\theta

Multiplying by

2

gives the identity.

cos(2θ) = 1 - 2 sin²θ — the law of cosines gives

|AB|^2 = 2 - 2\cos(2\theta)

, while the half-chord computation gives

|AB|^2 = 4\sin^2\theta

. Equating the two yields the result.

For full coverage of these proofs and equivalent forms, see the double angle identities theory page.

Derived Identities: tan(2θ), csc(2θ), sec(2θ), cot(2θ)

The four remaining identities follow directly from the first two:

tan(2θ) — from

\tan = \sin / \cos

:

\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}

csc(2θ) — reciprocal of

\sin(2\theta)

:

\csc(2\theta) = \frac{1}{2\sin\theta\cos\theta}

sec(2θ) — reciprocal of

\cos(2\theta)

:

\sec(2\theta) = \frac{1}{1 - 2\sin^2\theta}

cot(2θ) — reciprocal of

\tan(2\theta)

:

\cot(2\theta) = \frac{1 - \tan^2\theta}{2\tan\theta}

For step-by-step derivations of each, see the trigonometric identities page and the reciprocal identities page.

Why Double-Angle Identities Matter

Double-angle identities show up across mathematics and physics:

• Integration —

\sin^2\theta

and

\cos^2\theta

become integrable after substituting

\cos(2\theta) = 1 - 2\sin^2\theta

or

2\cos^2\theta - 1

.
• Equation solving — equations mixing

\sin\theta

with

\sin(2\theta)

collapse to single-angle equations after substitution.
• Wave physics and signal processing — sums of sinusoids reduce via these formulas, separating frequency components.
• Geometry and circular motion — relating arc, chord, and apothem in regular polygons uses

\sin(2\theta)

and

\cos(2\theta)

directly.

For applications and worked examples, see the trigonometric identities applications page.

Related Concepts and Tools

Continue exploring with these connected resources:

• Pythagorean Identities —

\sin^2\theta + \cos^2\theta = 1

and its companions.
• Sum and Difference Identities —

\sin(\alpha \pm \beta)

and

\cos(\alpha \pm \beta)

, from which double-angle identities follow as the case

\alpha = \beta

.
• Half-Angle Identities — solve the double-angle formulas backward to express

\sin(\theta/2)

and

\cos(\theta/2)

.
• Unit Circle — geometric setup for every identity in this tool.
• Trigonometric Functions Graphs — see how

\sin

,

\cos

,

\tan

and their reciprocals evolve as

\theta

varies.
• Triangle Explorer — interactive triangles with built-in law of sines and law of cosines.
• Negative Angle Identities — the parity rules that fix the sign of each formula at

-\theta

.
• Supplementary Angle Identities — how

\pi - \theta

interacts with the double-angle results.
• Basic Trigonometric Identities — the reciprocal and quotient identities used in the derived tan, csc, sec and cot forms.

The Sine Double-Angle Identity

The identity

\sin(2\theta) = 2\sin\theta\cos\theta

is proved geometrically in the explorer by computing the area of one isosceles triangle two different ways.

The complete sine proof, frozen

The area of one isosceles triangle computed two ways: ½ sin 2θ must equal sin θ cos θ.

The proof runs through six stages: setup, area the first way, bisect, read off the legs, area the second way, and equate.

This identity is the source of two others in the tool: cosecant's formula is its reciprocal, and tangent's numerator comes from it.

The Cosine Double-Angle Identity

The identity

\cos(2\theta) = 1 - 2\sin^2\theta

falls out of measuring the chord

AB

two ways — once by the law of cosines, once through the half-chords.

The complete cosine proof, frozen

The chord AB measured twice — law of cosines against half-chords — forcing cos 2θ = 1 − 2 sin²θ.

The proof's six stages: setup, law of cosines, bisect, read off the half-chord, square the chord, and equate.

Substituting

\sin^2\theta = 1 - \cos^2\theta

gives the equivalent forms

2\cos^2\theta - 1

and

\cos^2\theta - \sin^2\theta

. In the tool, secant's identity is this one inverted, and tangent's denominator comes from it.

The Tangent Double-Angle Identity

Tangent needs no new geometry: it is sine over cosine, so its double-angle formula follows algebraically from the two proved identities.

tan(2θ), derived

Three algebraic lines: sine over cosine, substitute both proved identities, divide through by cos²θ.

\tan(2\theta) = \frac{\sin(2\theta)}{\cos(2\theta)} = \frac{2\sin\theta\cos\theta}{\cos^2\theta - \sin^2\theta} = \frac{2\tan\theta}{1 - \tan^2\theta}

The last step divides numerator and denominator by

\cos^2\theta

— the move that turns a sine-and-cosine expression into a pure tangent one. The formula fails where

\tan\theta = \pm 1

(

\theta = 45°

), exactly where

2\theta = 90°

makes

\tan(2\theta)

undefined. Its own reciprocal gives cotangent's formula; its ingredients come from sine and cosine.

The Cosecant Double-Angle Identity

Cosecant is the reciprocal of sine, so its double-angle formula is one substitution away from the geometric result.

csc(2θ), derived

One substitution into the reciprocal: 1 over the sine identity.

\csc(2\theta) = \frac{1}{\sin(2\theta)} = \frac{1}{2\sin\theta\cos\theta}

The formula is undefined wherever

\sin(2\theta) = 0

— within the tool's

10°

–

80°

slider range that never happens, so the verification cards always agree. The source identity is sine's, reachable from the card's jump button.

The Secant Double-Angle Identity

Secant inverts cosine, so the double-angle version inverts the cosine identity.

sec(2θ), derived

The cosine identity inverted — with poles wherever cos 2θ = 0.

\sec(2\theta) = \frac{1}{\cos(2\theta)} = \frac{1}{1 - 2\sin^2\theta}

It diverges where

\cos(2\theta) = 0

, i.e. at

\theta = 45°

— sweep the slider there and watch both verification cards blow up together, which is itself a check that the two sides agree. The source identity is cosine's.

The Cotangent Double-Angle Identity

Cotangent is the reciprocal of tangent, so its formula is the tangent identity flipped upside down.

cot(2θ), derived

The tangent formula flipped upside down: zeros and poles exchange places.

\cot(2\theta) = \frac{1}{\tan(2\theta)} = \frac{1 - \tan^2\theta}{2\tan\theta}

Flipping exchanges the roles of the zeros and the poles: cotangent diverges where tangent is zero and vanishes where tangent diverges. Because tangent itself was derived, cotangent sits two steps from the geometry — resting ultimately on sine and cosine.

Sine Proof, Step 1: Setup

The sine proof opens with two radii

OA

and

OB

of length

1

meeting at the center

O

with angle

2\theta

between them. Together with the chord

AB

they form an isosceles triangle.

Step 1: the isosceles setup

Two unit radii meeting at 2θ, joined by the chord AB.

Everything the proof needs is already in this picture: a triangle whose apex angle is the double angle we want, built from sides whose length we know exactly.

Sine Proof, Step 2: Area, First Way

The triangle's area comes from the standard formula: half the product of two sides times the sine of the included angle. With

OA = OB = 1

meeting at

2\theta

:

Step 2: area, first way

Half the product of the unit sides times sin 2θ shades the whole triangle.

\text{area} = \tfrac{1}{2} \cdot 1 \cdot 1 \cdot \sin(2\theta) = \tfrac{1}{2}\sin(2\theta)

This is the left-hand side of the identity in disguise — one honest measurement of the shaded region.

Sine Proof, Step 3: Bisect

Drop

OM

perpendicular to the chord

AB

. Because the triangle is isosceles,

OM

bisects the apex: two half-angles of

\theta

at

O

, and two congruent right triangles.

Step 3: the bisector

OM splits the apex into two θ halves and makes two congruent right triangles.

The small square at

M

marks the right angle — the key that unlocks the next step, since right triangles are where

\sin\theta

and

\cos\theta

live as plain side lengths.

Sine Proof, Step 4: Read Off the Legs

In right triangle

OMA

the hypotenuse is

OA = 1

and the angle at

O

is

\theta

. Its legs are therefore exactly the basic ratios:

Step 4: legs as ratios

A unit hypotenuse means the legs are literally sin θ and cos θ.

MA = \sin\theta \qquad OM = \cos\theta

No approximation, no extra construction — with a unit hypotenuse, opposite and adjacent legs ARE sine and cosine.

Sine Proof, Step 5: Area, Second Way

Each right triangle has legs

\sin\theta

and

\cos\theta

, so each has area

\tfrac{1}{2}\sin\theta\cos\theta

. The two congruent halves together give:

Step 5: area, second way

Two congruent halves, each ½ sin θ cos θ — together sin θ cos θ.

\text{area} = 2 \cdot \tfrac{1}{2}\sin\theta\cos\theta = \sin\theta\cos\theta

The same shaded region as step 2, measured a second, independent way.

Sine Proof, Step 6: Equate

Two measurements of one area must agree:

Step 6: equate

The two area measurements meet: sin 2θ = 2 sin θ cos θ.

\tfrac{1}{2}\sin(2\theta) = \sin\theta\cos\theta \;\;\Longrightarrow\;\; \sin(2\theta) = 2\sin\theta\cos\theta

The verification cards below the scene confirm it numerically at every slider position — both sides always match to three decimals.

Cosine Proof, Step 1: Setup

The cosine proof starts from the same figure as sine's: radii

OA

and

OB

of length

1

with apex angle

2\theta

at the center.

Step 1: same setup, new target

The same unit triangle — but now the chord AB is the thing to measure.

This time the target of the measurement will not be the triangle's area but the length of the chord

AB

.

Cosine Proof, Step 2: Law of Cosines on Triangle OAB

Apply the law of cosines to the triangle, with the apex angle

2\theta

between the two unit sides:

Step 2: law of cosines

|AB|² = 2 − 2 cos 2θ, straight from the apex angle.

|AB|^2 = 1^2 + 1^2 - 2 \cdot 1 \cdot 1 \cdot \cos(2\theta) = 2 - 2\cos(2\theta)

The chord's squared length now contains

\cos(2\theta)

— the quantity the proof is hunting.

Cosine Proof, Step 3: Bisect

Drop the perpendicular bisector

OM

: it lands on the midpoint

M

of the chord and splits the apex into two half-angles of

\theta

.

Step 3: the bisector

OM lands on the midpoint M, halving the apex into two θ angles.

Same construction as in the sine proof, used here for a different purpose — to measure the chord instead of the area.

Cosine Proof, Step 4: Read Off the Half-Chord

Right triangle

OMA

has hypotenuse

1

and angle

\theta

at

O

, so the half-chord is

MA = \sin\theta

— and the full chord is twice that:

Step 4: the half-chord

MA = sin θ, so the whole chord is 2 sin θ.

AB = 2\sin\theta

A second, completely independent expression for the same chord the law of cosines measured.

Cosine Proof, Step 5: Square the Chord

Square the half-chord result to match the form of step 2:

Step 5: square it

(2 sin θ)² = 4 sin²θ — the second expression for |AB|².

|AB|^2 = (2\sin\theta)^2 = 4\sin^2\theta

Both routes now express

|AB|^2

— one through

\cos(2\theta)

, one through

\sin^2\theta

.

Cosine Proof, Step 6: Equate

Set the two expressions for

|AB|^2

equal and solve:

Step 6: equate

The two chord measurements force cos 2θ = 1 − 2 sin²θ.

2 - 2\cos(2\theta) = 4\sin^2\theta \;\;\Longrightarrow\;\; \cos(2\theta) = 1 - 2\sin^2\theta

As with sine, the verification cards keep both sides in numerical agreement across the whole slider range.

Double Angle Trigonometric Identities

Key Terms

Switching Between Functions

Adjusting the Angle θ

Playing Through a Geometric Proof

Reading the Geometric Scene

Working with Derived Identities

Reading the Formula Table

Verifying Identities Numerically

Geometric Proofs: sin(2θ) and cos(2θ)

Derived Identities: tan(2θ), csc(2θ), sec(2θ), cot(2θ)

Why Double-Angle Identities Matter

Related Concepts and Tools

The Sine Double-Angle Identity

The Cosine Double-Angle Identity

The Tangent Double-Angle Identity

The Cosecant Double-Angle Identity

The Secant Double-Angle Identity

The Cotangent Double-Angle Identity

Sine Proof, Step 1: Setup

Sine Proof, Step 2: Area, First Way

Sine Proof, Step 3: Bisect

Sine Proof, Step 4: Read Off the Legs

Sine Proof, Step 5: Area, Second Way

Sine Proof, Step 6: Equate

Cosine Proof, Step 1: Setup

Cosine Proof, Step 2: Law of Cosines on Triangle OAB

Cosine Proof, Step 3: Bisect

Cosine Proof, Step 4: Read Off the Half-Chord

Cosine Proof, Step 5: Square the Chord

Cosine Proof, Step 6: Equate