Sum and Difference Identities: Classroom Guide, Examples & Practice

Q: What is the fastest recognition cue for Sum and Difference Identities?

Look for **$\sin(A+B)$**, **exact value of $75°$ or $15°$**, **$A\pm B$ inside the function**, **combine two angles**, but treat those words as clues, not proof. A word problem can contain a familiar keyword and still ask for a different idea. After noticing the cue, ask the recognition question: Is the argument of the trig function a sum or difference of two angles I want to break apart? That question protects you from using a memorized procedure in the wrong place.

Section 1

Quick Answer

Sum and difference identities expand a trig function of $A\pm B$ into products of the two angles' sines and cosines. Use them to evaluate non-standard angles (like $75°=45°+30°$ ) or to simplify expressions hiding this product pattern. The cue is a trig function whose argument is itself a sum or difference of angles. Before calculating, ask: Is the argument of the trig function a sum or difference of two angles I want to break apart?

Section 2

Why This Matters

They unlock exact values for angles outside the special set and are the launchpad for double-angle and product-to-sum work. The fatal error — assuming $\sin(A+B)=\sin A+\sin B$ — wrecks all of trig, since trig functions are not linear. Recognizing it by "Is the argument of the trig function a sum or difference of two angles I want to break apart?" — rather than by familiar numbers — is what lets a student tell it apart from double-angle identities and pythagorean identities and distributing the function (wrong) in a mixed problem set.

Section 3

Intuitive Explanation

Two rotations stacked: turn by $A$ , then by $B$ ; the final point's coordinates mix both angles' sines and cosines rather than just adding them. This is the clean version of the idea because the visible structure matches the concept before any formula or procedure is chosen.

Distributing the function over the sum: $\sin(A+B)\ne\sin A+\sin B$ — the expansion mixes sines and cosines as cross products. That contrast matters because many wrong answers come from recognizing a surface feature, such as a familiar number or word, instead of the actual task.

A useful way to slow down is to name the signal words and then test them. Words like ** $\sin(A+B)$ **, **exact value of $75°$ or $15°$ **, ** $A\pm B$ inside the function**, **combine two angles**, ** $\cos(A-B)$ ** are helpful clues, but they are not enough by themselves. They must point to the same structure as the mental model: $\sin(A\pm B)$ and $\cos(A\pm B)$ expand into cross products of the separate angles' sines and cosines.

The recognition test is simple: Is the argument of the trig function a sum or difference of two angles I want to break apart? If yes, sum and difference identities is probably the right tool; if not, compare with Double-angle identities or Pythagorean identities or Distributing the function (wrong) before calculating.

Core idea

$\sin(A\pm B)$ and $\cos(A\pm B)$ expand into cross products of the separate angles' sines and cosines.

Section 4

When to Use

Use Sum and Difference Identities when the angle inside a trig function is written as a sum or difference of two angles you know. Strong signals include ** $\sin(A+B)$ **, **exact value of $75°$ or $15°$ **, ** $A\pm B$ inside the function**, **combine two angles**, ** $\cos(A-B)$ **. The safest workflow is to read the final question first, identify what kind of answer it wants, and then test the structure. Do not use sum and difference identities just because familiar numbers appear; first decide whether the situation answers "Is the argument of the trig function a sum or difference of two angles I want to break apart?" with yes.

✨ Pro tip

Ask: Is the argument of the trig function a sum or difference of two angles I want to break apart?

Section 5

How to Recognize It

Before using Sum and Difference Identities, check the structure of the problem, not just the vocabulary. These questions force the same recognition move from several angles: the task, the signal words, the nearest confusion, and the thing that would make the concept fail.

Is the argument of the trig function a sum or difference of two angles I want to break apart?

If yes, the problem matches sum and difference identities. If no, pause before applying the procedure, because the same numbers may belong to a different idea.
Which words signal the structure?

Look for $\sin(A+B)$ , exact value of $75°$ or $15°$ , $A\pm B$ inside the function, combine two angles. These words are useful only after the situation matches them; a keyword without structure is not proof.
What is the nearest confusion?

Double-angle identities is the common trap here: The special case where both angles are equal, $A=B$ . Compare the desired final answer before choosing a method.
What answer form should I expect?

The answer should fit this mental model: $\sin(A\pm B)$ and $\cos(A\pm B)$ expand into cross products of the separate angles' sines and cosines. If the expected answer sounds more like double-angle identities, use the comparison table before solving.
What would make this NOT Sum and Difference Identities?

Distributing the function over the sum: $\sin(A+B)\ne\sin A+\sin B$ — the expansion mixes sines and cosines as cross products. This tells you when to switch tools instead of forcing the concept.

Section 6

Sum and Difference Identities vs Common Confusions

The hard part is recognizing when the task is really about sum and difference identities instead of a nearby idea. Read the final answer the problem wants, then ask which row describes the structure before you start calculating.

Sum and Difference Identities vs Common Confusions
Type	Meaning	Key test	Formula	Example
Sum and Difference Identities	Use this when the angle inside a trig function is written as a sum or difference of two angles you know. The deciding question is: Is the argument of the trig function a sum or difference of two angles I want to break apart?	Is the argument of the trig function a sum or difference of two angles I want to break apart?	$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$ $\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$ $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$	Find the exact value of $\cos 75°$ .
Double-angle identities	The special case where both angles are equal, $A=B$ .	Use when the angle is exactly twice another, like $2\theta$.	$\sin 2\theta=2\sin\theta\cos\theta$	$\sin 80°$ from $40°$
Pythagorean identities	Relate squares of one angle's trig values; no second angle involved.	Use when you have $\sin^2$ and $\cos^2$ of the same angle.	$\sin^2\theta+\cos^2\theta=1$	Simplify $1-\cos^2\theta$
Distributing the function (wrong)	The illegal move of splitting $\sin(A+B)$ into $\sin A+\sin B$ .	Never use — trig functions are not linear; this is always wrong.		$\sin(30°+60°)\ne\sin30°+\sin60°$

Sum and Difference Identities

Meaning: Use this when the angle inside a trig function is written as a sum or difference of two angles you know. The deciding question is: Is the argument of the trig function a sum or difference of two angles I want to break apart?
Key test: Is the argument of the trig function a sum or difference of two angles I want to break apart?
Formula: $\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$
$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$
$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$
Example: Find the exact value of $\cos 75°$ .

Double-angle identities

Meaning: The special case where both angles are equal, $A=B$ .
Key test: Use when the angle is exactly twice another, like $2\theta$.
Formula: $\sin 2\theta=2\sin\theta\cos\theta$
Example: $\sin 80°$ from $40°$

Pythagorean identities

Meaning: Relate squares of one angle's trig values; no second angle involved.
Key test: Use when you have $\sin^2$ and $\cos^2$ of the same angle.
Formula: $\sin^2\theta+\cos^2\theta=1$
Example: Simplify $1-\cos^2\theta$

Distributing the function (wrong)

Meaning: The illegal move of splitting $\sin(A+B)$ into $\sin A+\sin B$ .
Key test: Never use — trig functions are not linear; this is always wrong.
Example: $\sin(30°+60°)\ne\sin30°+\sin60°$

Section 7

Formula & Notation

\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B

\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B

\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}

\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B

;

\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B

;

\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}

How to read it: Note the sign pattern: in the cosine formula, $\pm$ becomes $\mp$ (signs are opposite). In sine and tangent, signs match.

Section 8

Worked Examples

Example 1 — Exact value of cos 75°

Easy

Problem

Find the exact value of $\cos 75°$ .

Solution

$75°=45°+30°$ , a sum of two known angles.

Name the structure before touching arithmetic — that is what makes the right method obvious.
Ask the recognition question: Is the argument of the trig function a sum or difference of two angles I want to break apart?

If the answer is yes, the concept applies; the cue, not a keyword, decides the method.
Apply $\cos(A+B)=\cos A\cos B-\sin A\sin B$ with $A=45°,B=30°$ .

The rule is chosen only after the structure matches, so the steps mean something.
$\frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2}-\frac{\sqrt2}{2}\cdot\frac{1}{2}=\frac{\sqrt6-\sqrt2}{4}$ .

Keep units, shape, or answer form tied to the story so the work does not become symbol pushing.
Check the answer against the original question.

It should fit the mental model — trig of a sum is a mix, never a sum of trigs. If it does not, revisit the recognition step before changing the arithmetic.

Answer

$\frac{\sqrt6-\sqrt2}{4}$

Takeaway: Splitting an angle into a known sum lets the identity build an exact value.

Example 2 — Both angles are equal

Standard

Problem

Find $\cos 2\theta$ given $\theta$ — do you use the difference identity?

Solution

Notice why this looks like the same concept.

Nearby language or numbers can tempt you toward trig of a sum is a mix, never a sum of trigs.
The two angles are identical, $A=B=\theta$ , not distinct.

Spotting what actually changed is what separates this from the concept it resembles.
Collapse the sum identity into the double-angle form instead of two separate angles.

The nearby idea may share numbers but answers a different question, so it needs a different move.
State the result in the language of the actual task.

$\cos 2\theta=\cos^2\theta-\sin^2\theta$ . Name it for what the problem really asked, not the concept you first expected.
Say the contrast in one sentence.

Equal angles specialize the sum identity into the double-angle one.

Answer

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

Takeaway: Equal angles specialize the sum identity into the double-angle one.

Example 3 — Spot the trap: Trig of a sum is a mix, never a sum of trigs

Application

Problem

A student starts with this idea: "Splitting $\cos(A+B)$ as $\cos A+\cos B$ " What should they check before accepting that reasoning?

Solution

Pause before the first move.

The first move is a decision, not a calculation — does the situation really match trig of a sum is a mix, never a sum of trigs.
Run the recognition test: Is the argument of the trig function a sum or difference of two angles I want to break apart?

This is the single check that the trap skips.
it expands to $\cos A\cos B-\sin A\sin B$ , a product mix.

Stating the safer rule turns the mistake into a checkable step instead of a vague "be careful."
Compare with the nearest confusion, Double-angle identities.

The special case where both angles are equal, $A=B$ .
State the corrected decision and reuse it.

Using the concept only when the structure matches leaves a process the student can repeat on a new problem.

Answer

it expands to $\cos A\cos B-\sin A\sin B$ , a product mix.

Takeaway: The recognition step prevents the common trap: Splitting $\cos(A+B)$ as $\cos A+\cos B$

Section 9

Common Mistakes

Common slip-up

Splitting $\cos(A+B)$ as $\cos A+\cos B$

The right idea

it expands to $\cos A\cos B-\sin A\sin B$ , a product mix.

Common slip-up

Getting the cosine sign wrong

The right idea

in cosine the inside $+$ becomes a $-$ between the products ( $\mp$ flips).

Common slip-up

Using matching signs for cosine

The right idea

sine and tangent keep the sign, cosine reverses it.

Section 10

Mini Practice

Try these on your own. Tap Reveal when you want to check.

What clue tells you this is a Sum and Difference Identities situation: Find the exact value of $\cos 75°$ .

Hint: Is the argument of the trig function a sum or difference of two angles I want to break apart?
Find the exact value of $\cos 75°$ .

Hint: Apply $\cos(A+B)=\cos A\cos B-\sin A\sin B$ with $A=45°,B=30°$ .
Why is this a contrast case instead of Sum and Difference Identities: Find $\cos 2\theta$ given $\theta$ — do you use the difference identity?

Hint: The two angles are identical, $A=B=\theta$ , not distinct.
Fix this thinking: Splitting $\cos(A+B)$ as $\cos A+\cos B$

Hint: Name the recognition cue before choosing a rule.
Which is the better fit here: Sum and Difference Identities or Double-angle identities? Explain the deciding difference.

Hint: For Sum and Difference Identities, ask: Is the argument of the trig function a sum or difference of two angles I want to break apart?
Write one sentence that would remind a classmate how to recognize Sum and Difference Identities.

Hint: Use the mental model "Trig of a sum is a mix, never a sum of trigs." and one signal word.

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Section 11

Frequently Asked Questions

How do I know when to use Sum and Difference Identities?

Use Sum and Difference Identities when the angle inside a trig function is written as a sum or difference of two angles you know. Do not start from the numbers alone; first name the structure of the situation. The fastest check is: Is the argument of the trig function a sum or difference of two angles I want to break apart? If the answer is yes and the wording matches cues like $\sin(A+B)$ , exact value of $75°$ or $15°$ , $A\pm B$ inside the function, then sum and difference identities is probably the right tool.

What is Sum and Difference Identities most often confused with?

Sum and Difference Identities is often confused with Double-angle identities. Double-angle identities means The special case where both angles are equal, $A=B$ . The difference is not just vocabulary; it changes the action you take. For sum and difference identities, the key test is "Is the argument of the trig function a sum or difference of two angles I want to break apart?" For double-angle identities, the better cue is: Use when the angle is exactly twice another, like $2\theta$ .

What is the fastest recognition cue for Sum and Difference Identities?

Look for $\sin(A+B)$ , exact value of $75°$ or $15°$ , $A\pm B$ inside the function, combine two angles, but treat those words as clues, not proof. A word problem can contain a familiar keyword and still ask for a different idea. After noticing the cue, ask the recognition question: Is the argument of the trig function a sum or difference of two angles I want to break apart? That question protects you from using a memorized procedure in the wrong place.

What mistake should I avoid with Sum and Difference Identities?

Avoid this thinking: "Splitting $\cos(A+B)$ as $\cos A+\cos B$ " That mistake usually happens when the student jumps to a rule before checking the situation. The safer version is: it expands to $\cos A\cos B-\sin A\sin B$ , a product mix. A good habit is to say the mental model out loud first: "Trig of a sum is a mix, never a sum of trigs." Then choose the calculation or representation.

How can I tell this apart from Pythagorean identities?

Pythagorean identities is the better fit when the task is about this: Relate squares of one angle's trig values; no second angle involved. Sum and Difference Identities is the better fit when the angle inside a trig function is written as a sum or difference of two angles you know. If both ideas seem possible, compare what the problem wants as the final answer. The desired output often reveals whether you should use sum and difference identities or switch to the nearby concept.

Why does Sum and Difference Identities matter?

They unlock exact values for angles outside the special set and are the launchpad for double-angle and product-to-sum work. The fatal error — assuming $\sin(A+B)=\sin A+\sin B$ — wrecks all of trig, since trig functions are not linear. The practical value is recognition: once you can spot sum and difference identities, you can choose a method before calculating. That makes later topics easier because you are not memorizing isolated tricks; you are recognizing the same structure when it appears in a new representation.

Section 12

Learning Path

← Before

Pythagorean Trigonometric Identities Trigonometric Functions

Sum and Difference Identities

You are here

Next →

Double-Angle Identities

Before this, students should be comfortable with Pythagorean Trigonometric Identities and Trigonometric Functions. This page focuses on the recognition cue: Is the argument of the trig function a sum or difference of two angles I want to break apart? That cue is the bridge between earlier skills and later problem solving: students first learn to identify the structure, then they learn which calculation, diagram, graph, or proof move belongs to it. After this, Double-Angle Identities become easier to recognize.

Section 13