Proving Angle Addition/Subtraction Formulas - Precalculus

SECTION 1

Historical Context & Motivation

Long before calculators existed, astronomers and navigators needed ways to compute trigonometric values for angles that didn't appear in standard tables. The angle addition and subtraction formulas arose from this practical need: how do you find sin 75° if you only know sine and cosine for 45° and 30°? The answer lies in breaking a complicated angle into the sum or difference of simpler ones. This idea stretches back thousands of years and remains one of the most powerful tools in trigonometry.

~150 CE • Ptolemy's Chord Tables

The Greek astronomer Claudius Ptolemy developed a chord-subtraction formula in his Almagest, which is equivalent to our modern cosine subtraction formula, to build astronomical tables.

~500 CE • Indian Mathematicians

Scholars like Aryabhata and later Bhaskara I refined sine tables using sum-and-difference relationships, expressing results in the sine function rather than Ptolemy's chord function.

~1400 • Islamic Golden Age Refinement

Mathematicians such as al-Kāshī produced highly accurate trig tables, relying on angle addition identities to interpolate between known values.

1748 • Euler's Analytic Approach

Leonhard Euler connected the addition formulas to complex exponentials through his famous identity e^iθ = cos θ + i sin θ, giving the formulas an elegant algebraic foundation.

Today • Modern Applications

The addition formulas are essential in signal processing, physics (wave interference), engineering, and computer graphics — anywhere angles combine.

The central question these formulas answer is deceptively simple: if you know the sine and cosine of two individual angles α and β, can you find the exact sine, cosine, and tangent of their sum α + β or difference α − β without measuring the combined angle directly? The answer is a definitive yes, and proving why these formulas work is the focus of this lesson.

SECTION 2

Core Principles & Definitions

Before diving into proofs, you need a firm grasp of a few foundational ideas. These principles tie together the unit circle, distance formula, and the special relationship between sine and cosine that makes the addition formulas possible.

Unit Circle Coordinates

Any angle θ measured from the positive x-axis determines a point on the unit circle with coordinates (cos θ, sin θ). This is the definition that connects algebra and geometry.

The Pythagorean Identity

Because (cos θ, sin θ) lies on a circle of radius 1, we always have cos²θ + sin²θ = 1. This identity appears repeatedly in the proofs.

The Distance Formula

The distance between two points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²]. We will use this on two pairs of points on the unit circle to derive the cosine subtraction formula.

Cofunction & Even/Odd Identities

Cosine is an even function: cos(−θ) = cos θ. Sine is odd: sin(−θ) = −sin θ. The cofunction identities relate sine to cosine: sin θ = cos(π/2 − θ).

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KEY TAKEAWAY

Think of the addition formulas like a recipe: you already know the "ingredients" (sin α, cos α, sin β, cos β), and the formula tells you how to combine them to "cook" the trig value of the new angle α + β. You never need to measure the combined angle directly — just mix the parts you already know.

SECTION 3

Visual Proof: The Unit Circle Approach

The most intuitive proof of the cosine subtraction formula uses the unit circle and the distance formula. The diagram below shows two configurations of the same arc on the unit circle. By rotating the arc so that one endpoint sits on the positive x-axis, we can set up two distance expressions that must be equal — and expanding them reveals the formula.

In Configuration 1, the chord connects points P and Q on the unit circle, separated by the arc from angle α to angle β. In Configuration 2, the same arc is rotated so one endpoint sits at A = (1, 0) and the other at B = (cos(β − α), sin(β − α)). Since rotating an arc doesn't change its chord length, d₁ = d₂. Squaring both distances and simplifying produces the cosine subtraction formula.

Here is the algebra. In Configuration 1, use the distance formula between P = (cos α, sin α) and Q = (cos β, sin β): d₁² = (cos β − cos α)² + (sin β − sin α)². Expand this to get cos²β − 2 cos α cos β + cos²α + sin²β − 2 sin α sin β + sin²α. Apply the Pythagorean identity (cos²θ + sin²θ = 1) twice to simplify: d₁² = 2 − 2(cos α cos β + sin α sin β).

In Configuration 2, the distance from A = (1, 0) to B = (cos(β − α), sin(β − α)) gives d₂² = (cos(β − α) − 1)² + (sin(β − α))². Expanding: d₂² = cos²(β − α) − 2 cos(β − α) + 1 + sin²(β − α) = 2 − 2 cos(β − α). Setting d₁² = d₂² yields the cosine subtraction formula: cos(β − α) = cos α cos β + sin α sin β.

SECTION 4

Mathematical Framework: All Six Formulas

Once you have the cosine subtraction formula, every other addition/subtraction formula can be derived from it using only the even/odd identities and the cofunction identities. Let's build them one by one.

COSINE SUBTRACTION

cos(α − β) = cos α cos β + sin α sin β

Proved above via the unit circle distance argument. This is the master identity from which all others follow.

COSINE ADDITION

cos(α + β) = cos α cos β − sin α sin β

Replace β with −β in the subtraction formula. Since cos(−β) = cos β and sin(−β) = −sin β, the + sign flips to −.

SINE SUBTRACTION

sin(α − β) = sin α cos β − cos α sin β

Use the cofunction identity sin θ = cos(π/2 − θ). Write sin(α − β) = cos(π/2 − (α − β)) = cos((π/2 − α) + β), then apply the cosine addition formula.

SINE ADDITION

sin(α + β) = sin α cos β + cos α sin β

Replace β with −β in the sine subtraction formula. The − changes to + because sin(−β) = −sin β introduces a double negative.

Tangent Formulas

The tangent addition and subtraction formulas follow from dividing the sine formula by the cosine formula, since tan θ = sin θ / cos θ. After dividing both numerator and denominator by cos α cos β, you get a clean result.

TANGENT ADDITION

tan(α + β) = (tan α + tan β) / (1 − tan α tan β)

Valid whenever cos α, cos β, and cos(α + β) are all nonzero.

TANGENT SUBTRACTION

tan(α − β) = (tan α − tan β) / (1 + tan α tan β)

Replace β with −β in the addition formula. Since tan(−β) = −tan β, the signs adjust accordingly.

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Sign Pattern Tip

Notice the sign pattern: in the cosine formulas, the sign in front of the product switches — subtraction gives +, addition gives −. In the sine formulas, the sign matches the operation. Many students remember this with the phrase: "cosine is contrary, sine is the same."

SECTION 5

Step-by-Step Derivation Map

The diagram below illustrates the logical flow from the single master identity — cos(α − β) — to all six addition/subtraction formulas. Each arrow represents one algebraic substitution.

The derivation map shows that cos(α − β) is the single starting point. Replacing β with −β yields the cosine addition formula. Applying the cofunction identity yields the sine subtraction formula, and replacing β again gives sine addition. Dividing sine by cosine produces the tangent formulas.

Formula	Expression	Key Derivation Step
cos(α − β)	cos α cos β + sin α sin β	Unit circle distance proof (master identity)
cos(α + β)	cos α cos β − sin α sin β	Substitute −β for β; use cos(−β) = cos β, sin(−β) = −sin β
sin(α − β)	sin α cos β − cos α sin β	Cofunction: sin θ = cos(π/2 − θ)
sin(α + β)	sin α cos β + cos α sin β	Substitute −β for β in sine subtraction
tan(α + β)	(tan α + tan β) / (1 − tan α tan β)	Divide sin(α + β) by cos(α + β); simplify
tan(α − β)	(tan α − tan β) / (1 + tan α tan β)	Substitute −β for β in tangent addition

Complete Summary of the Six Angle Addition/Subtraction Formulas

SECTION 6

Worked Example: Finding sin 75° Exactly

Let's put the sine addition formula to work. We want to find the exact value of sin 75°. Notice that 75° = 45° + 30°, and we know exact values for sine and cosine of both 45° and 30°.

Find the Exact Value of sin 75°

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Step 1 — Decompose the Angle

Write 75° as the sum of two special angles: 75° = 45° + 30°.

α = 45°, β = 30°
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Step 2 — Write the Sine Addition Formula

Apply sin(α + β) = sin α cos β + cos α sin β. Substituting: sin 75° = sin 45° cos 30° + cos 45° sin 30°.
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Step 3 — Substitute Known Values

Recall the special angle values: sin 45° = √2/2, cos 45° = √2/2, sin 30° = 1/2, cos 30° = √3/2. Substitute: sin 75° = (√2/2)(√3/2) + (√2/2)(1/2).
4
Step 4 — Multiply

Multiply each pair of fractions: sin 75° = √6/4 + √2/4.
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Step 5 — Combine

Since both fractions have denominator 4, combine them into a single fraction.

sin 75° = (√6 + √2) / 4 ≈ 0.9659

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Verification

You can check this on your calculator: sin 75° ≈ 0.9659, and (√6 + √2)/4 ≈ (2.449 + 1.414)/4 ≈ 3.863/4 ≈ 0.9659. They match!

SECTION 7

Common Mistakes & How to Avoid Them

When working with angle addition formulas, a few errors come up again and again. The table below catalogs the most common mistakes alongside correct approaches, so you can learn to spot them before they trip you up.

Common Mistake	Why It's Wrong	Correct Approach
sin(α + β) = sin α + sin β	Sine is NOT a linear function. You cannot distribute it over addition like multiplication.	Use sin(α + β) = sin α cos β + cos α sin β.
Mixing up signs in the cosine formula	"Cosine is contrary": the sign between terms is opposite to the operation. cos(A + B) uses minus, not plus.	cos(A + B) = cos A cos B − sin A sin B. Remember: addition → minus.
Forgetting to use exact special-angle values	Decimal approximations lose exactness and defeat the purpose of these formulas.	Keep radicals: sin 45° = √2/2, cos 30° = √3/2, etc.
Dividing wrong when deriving tangent	You must divide BOTH numerator and denominator by cos α cos β, not just the top.	tan(α + β) = sin(α + β)/cos(α + β), then divide every term by cos α cos β.

Pitfalls and Fixes for Angle Addition/Subtraction Formulas

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REMEMBER

The biggest single mistake students make is treating trig functions like they are linear: sin(A + B) ≠ sin A + sin B. Think of it like this — if you raise the temperature of an oven by combining two heat settings, the result depends on how they interact, not just their raw sum. The addition formulas capture that interaction.

SECTION 8

Connection to Advanced Identities

The angle addition formulas are not an isolated topic — they are the foundation for a whole family of more advanced trigonometric identities that you'll encounter in later courses. Understanding this connection shows you why mastering these formulas now pays dividends later.

Addition Formulas (This Lesson)	Advanced Identity	How It Connects
sin(α + β) = sin α cos β + cos α sin β	Double-Angle: sin 2θ = 2 sin θ cos θ	Set α = β = θ in the sine addition formula
cos(α + β) = cos α cos β − sin α sin β	Double-Angle: cos 2θ = cos²θ − sin²θ	Set α = β = θ in the cosine addition formula
cos(α − β) and cos(α + β)	Product-to-Sum: cos α cos β = ½[cos(α − β) + cos(α + β)]	Add the two cosine formulas and solve for the product
cos 2θ = 1 − 2 sin²θ	Half-Angle: sin²(θ/2) = (1 − cos θ)/2	Solve the double-angle formula for sin²θ, then replace θ with θ/2
Euler: e^iθ = cos θ + i sin θ	Complex Exponentials	Multiplying e^iα × e^iβ = e^i(α+β) reproduces both addition formulas at once

How Addition Formulas Extend to Advanced Identities

In AP Calculus, you'll use these identities to differentiate and integrate trigonometric functions. In physics, the addition formulas describe how waves combine — the principle of superposition depends on exactly these relationships. If you continue to engineering or computer science, signal processing relies on them daily. Mastering the proofs now gives you a deep understanding that memorization alone can't provide.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain in your own words why sin(α + β) ≠ sin α + sin β. Give a specific numerical example using α = 60° and β = 30° to illustrate.

Show answer

Sine is not a linear function, so it does not distribute over addition. Using the example: sin(60° + 30°) = sin 90° = 1. However, sin 60° + sin 30° = √3/2 + 1/2 ≈ 0.866 + 0.5 = 1.366, which does not equal 1. The addition formula accounts for how the two angles interact through cross-terms (sin α cos β and cos α sin β), which is why the correct formula sin(α + β) = sin α cos β + cos α sin β is necessary.

Question 2: BASIC CALCULATION

Use the cosine addition formula to find the exact value of cos 105°. (Hint: 105° = 60° + 45°.)

Show answer

Apply cos(α + β) = cos α cos β − sin α sin β with α = 60° and β = 45°. Substituting: cos 105° = cos 60° cos 45° − sin 60° sin 45° = (1/2)(√2/2) − (√3/2)(√2/2) = √2/4 − √6/4 = (√2 − √6)/4. This gives approximately (1.414 − 2.449)/4 ≈ −0.2588, which matches the calculator value of cos 105°.

Question 3: INTERMEDIATE

Find the exact value of tan 15° using the tangent subtraction formula. (Hint: 15° = 45° − 30°.)

Show answer

Apply tan(α − β) = (tan α − tan β)/(1 + tan α tan β) with α = 45° and β = 30°. Since tan 45° = 1 and tan 30° = 1/√3 = √3/3, substitute: tan 15° = (1 − √3/3)/(1 + 1 · √3/3) = (3/3 − √3/3)/(3/3 + √3/3) = (3 − √3)/(3 + √3). Rationalize by multiplying numerator and denominator by (3 − √3): [(3 − √3)²]/[(9 − 3)] = (9 − 6√3 + 3)/6 = (12 − 6√3)/6 = 2 − √3 ≈ 0.2679. This matches the known value of tan 15°.

Question 4: APPLIED

A surveyor measures angles of elevation of 40° and 25° from two different positions to the top of a building. She needs to compute sin(40° − 25°) = sin 15° to complete her distance calculation. Using the sine subtraction formula and the values sin 40° ≈ 0.6428, cos 40° ≈ 0.7660, sin 25° ≈ 0.4226, cos 25° ≈ 0.9063, verify that sin 15° ≈ 0.2588.

Show answer

Apply sin(α − β) = sin α cos β − cos α sin β with α = 40° and β = 25°. Substituting the given values: sin 15° = (0.6428)(0.9063) − (0.7660)(0.4226) = 0.5826 − 0.3237 = 0.2589. This rounds to 0.2588, matching the known value of sin 15°. The small rounding difference (0.0001) comes from the approximate decimal inputs. In practice, surveyors use these formulas to decompose angle differences when direct measurement of the difference is impractical.

Question 5: CRITICAL THINKING

Prove the sine addition formula sin(α + β) = sin α cos β + cos α sin β starting from the cosine subtraction formula cos(α − β) = cos α cos β + sin α sin β. You may use the cofunction identity sin θ = cos(π/2 − θ) and the even/odd identities. Show every step.

Show answer

Step 1: Write sin(α + β) using the cofunction identity: sin(α + β) = cos(π/2 − (α + β)) = cos((π/2 − α) − β). Step 2: Apply the cosine subtraction formula with the substitution A = π/2 − α and B = β: cos(A − B) = cos A cos B + sin A sin B = cos(π/2 − α) cos β + sin(π/2 − α) sin β. Step 3: Apply cofunction identities: cos(π/2 − α) = sin α and sin(π/2 − α) = cos α. Step 4: Substitute to get sin(α + β) = sin α cos β + cos α sin β. This completes the proof. Every step relies on previously established identities — the cosine subtraction formula (proved via the unit circle) and the cofunction identities (which follow from the symmetry of the unit circle).

SUMMARY

Lesson Summary

Summary

In this lesson you learned how to prove and apply the angle addition and subtraction formulas for sine, cosine, and tangent. The proof begins with the cosine subtraction formula, derived by comparing chord lengths on the unit circle using the distance formula. From this single master identity, every other formula follows through systematic substitutions: replacing β with −β using even/odd identities, applying cofunction identities to move between cosine and sine, and dividing to derive the tangent versions.

The six formulas are: cos(α ± β) = cos α cos β ∓ sin α sin β, sin(α ± β) = sin α cos β ± cos α sin β, and tan(α ± β) = (tan α ± tan β)/(1 ∓ tan α tan β). These identities let you compute exact trig values for non-standard angles like 15°, 75°, and 105° by decomposing them into sums or differences of special angles (30°, 45°, 60°). They also serve as the foundation for double-angle, half-angle, and product-to-sum identities that you'll encounter in more advanced courses.