# Addition Formulae

> Edexcel International GCSE Further Pure Mathematics · 4PM1
> Source: https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s10-addition-formulae/

This guide teaches you to use the given sin(A±B), cos(A±B) and tan(A±B) addition formulae to expand, simplify trigonometric expressions, and derive required double-angle identities for your Edexcel 4PM1 exam.

**Prerequisites:** [Knowledge of exact trig values for 30°, 45°, 60°](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s10-exact-trig-values/); [Understanding of basic trig identities (sin²x + cos²x = 1, tanx = sinx/cosx)](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s10-basic-trig-identities/)

## Learning objectives

- Expand and simplify trigonometric expressions using the given sin(A±B), cos(A±B) and tan(A±B) formulae
- Derive all double-angle identities directly from the addition formulae
- Calculate exact trigonometric values for non-standard angles using addition formulae
- Verify simple trigonometric identities using addition and double-angle rules

## Given Addition Formulae and Basic Application

All core addition formulae are provided on your Edexcel 4PM1 formula sheet, so you do not need to memorise them. You will use them to expand expressions with combined angles, calculate exact values of non-standard angles, and simplify complex trig statements.

**Addition Formulae (Given)** — The six core provided formulae are:
- $\sin(A + B) = \sin A \cos B + \cos A \sin B$
- $\sin(A - B) = \sin A \cos B - \cos A \sin B$
- $\cos(A + B) = \cos A \cos B - \sin A \sin B$
- $\cos(A - B) = \cos A \cos B + \sin A \sin B$
- $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
- $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

**Worked example:** Calculate the exact value of $\sin 75^\circ$ using the addition formula for $\sin(A+B)$.

1. Step 1: Express 75° as the sum of two angles with known exact values: $75^\circ = 45^\circ + 30^\circ$, so $A=45^\circ$, $B=30^\circ$
2. Step 2: Substitute into the given $\sin(A+B)$ formula:
3. $$\sin(45^\circ + 30^\circ) = \sin45^\circ \cos30^\circ + \cos45^\circ \sin30^\circ$$
4. Step 3: Substitute exact values: $\sin45^\circ = \frac{\sqrt{2}}{2}$, $\cos30^\circ = \frac{\sqrt{3}}{2}$, $\cos45^\circ=\frac{\sqrt{2}}{2}$, $\sin30^\circ=\frac{1}{2}$
5. $$= \left(\frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2} \times \frac{1}{2}\right) = \frac{\sqrt{6} + \sqrt{2}}{4}$$

> **Exam tip:** Always use exact values from 30/45/60 angles when asked for exact results, never use decimal approximations.

*Calculator:* allowed

## Deriving Double-Angle Identities from Addition Formulae

Double-angle identities are not given on the formula sheet, but you can derive them quickly by setting $B = A$ in the addition formulae. These identities are extremely useful for simplifying expressions with angles multiplied by 2.

> **tip**
>
> Memorise the three forms of $\cos 2A$ to save time in exams, as they are used frequently for simplification.

**Double-Angle Identities (Derived)** — You must be able to derive all of these from the given addition formulae:
- $\sin 2A = 2 \sin A \cos A$
- $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$
- $\tan 2A = \frac{2 \tan A}{1 - \tan^2 A}$

**Worked example:** Derive the identity for $\cos 2A$ in terms of $\cos^2 A$ only, starting from the $\cos(A+B)$ addition formula.

1. Step 1: Start with the given $\cos(A+B)$ formula:
2. $$\cos(A+B) = \cos A \cos B - \sin A \sin B$$
3. Step 2: Set $B = A$ to get the double angle:
4. $$\cos(A+A) = \cos A \cos A - \sin A \sin A \implies \cos 2A = \cos^2 A - \sin^2 A$$
5. Step 3: Use the Pythagorean identity $\sin^2 A = 1 - \cos^2 A$ to substitute:
6. $$\cos 2A = \cos^2 A - (1 - \cos^2 A) = 2\cos^2 A - 1$$

*Calculator:* allowed

## Expanding and Simplifying Trigonometric Expressions

A common exam question asks you to expand or simplify combined trig expressions using the addition formulae, often followed by simplification using basic identities or double-angle rules. Always show every step of your working to get full marks.

**Worked example:** Simplify the expression $\cos(\theta + 30^\circ) + \sin(\theta - 60^\circ)$, leaving your answer in its simplest form.

1. Step 1: Expand both terms using the addition formulae:
2. $$\cos(\theta + 30^\circ) = \cos\theta \cos30^\circ - \sin\theta \sin30^\circ = \frac{\sqrt{3}}{2}\cos\theta - \frac{1}{2}\sin\theta$$
3. $$\sin(\theta - 60^\circ) = \sin\theta \cos60^\circ - \cos\theta \sin60^\circ = \frac{1}{2}\sin\theta - \frac{\sqrt{3}}{2}\cos\theta$$
4. Step 2: Add the two expanded expressions together:
5. $$\left(\frac{\sqrt{3}}{2}\cos\theta - \frac{1}{2}\sin\theta\right) + \left(\frac{1}{2}\sin\theta - \frac{\sqrt{3}}{2}\cos\theta\right) = 0$$

> **Exam tip:** When simplifying, always cross out equal terms with opposite signs first to reduce the number of terms you need to work with.

*Calculator:* allowed

## Verifying Trigonometric Identities

You may be asked to show that two trigonometric expressions are equal using the addition formulae. Always start with the more complex side of the identity and manipulate it to match the simpler side.

**Worked example:** Show that $\sin(A + B) + \sin(A - B) = 2 \sin A \cos B$.

1. Step 1: Start with the left-hand side (LHS), which is the more complex side.
2. Step 2: Expand both $\sin(A+B)$ and $\sin(A-B)$ using the given addition formulae:
3. $$LHS = [\sin A \cos B + \cos A \sin B] + [\sin A \cos B - \cos A \sin B]$$
4. Step 3: Simplify by combining like terms:
5. $$LHS = \sin A \cos B + \cos A \sin B + \sin A \cos B - \cos A \sin B = 2 \sin A \cos B$$
6. This matches the right-hand side (RHS), so the identity is proven.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Mixing up the sign in the $\cos(A \pm B)$ formula
  - Why it fails: The sign in the cos expansion is the opposite of the operation between the angles, so students often write + instead of -
  - Correct: Remember the mnemonic: cos cross, sign flips: for $\cos(A+B)$ you subtract the product of sines, for $\cos(A-B)$ you add it.
- **Wrong:** Forgetting to derive double-angle identities if asked explicitly
  - Why it fails: Double-angle identities are not given on the formula sheet, so stating them without derivation will lose marks if the question asks for a derivation from addition rules
  - Correct: Always set $B=A$ in the relevant addition formula and show every step of the derivation when requested.
- **Wrong:** Using $\tan(A\pm B)$ formula when $A$ or $B$ is an odd multiple of $90^\circ$
  - Why it fails: Tangent is undefined at odd multiples of $90^\circ$, so the formula will produce an invalid result
  - Correct: Use sine and cosine addition formulae instead for angles where $\tan$ is undefined.
- **Wrong:** Incorrectly carrying signs when expanding $\sin(A-B)$ or $\cos(A-B)$
  - Why it fails: Students often forget to apply the negative sign to all terms in the second half of the expansion
  - Correct: Write the angle as $A + (-B)$ and use the standard addition formula to preserve sign consistency if you struggle with the subtraction form.
- **Wrong:** Using degree mode on your calculator when angles are given in radians, or vice versa
  - Why it fails: This will produce incorrect numerical values and invalidate your working
  - Correct: Always check the angle unit given in the question, and adjust your calculator mode accordingly if you are using a calculator for calculation.

## Cheatsheet

| Formula Type | Given / Derived | Identity |
| --- | --- | --- |
| Addition (Sine) | Given | $\sin(A\pm B) = \sin A \cos B \pm \cos A \sin B$ |
| Addition (Cosine) | Given | $\cos(A\pm B) = \cos A \cos B \mp \sin A \sin B$ |
| Addition (Tangent) | Given | $\tan(A\pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ |
| Double Angle (Sine) | Derived | $\sin2A = 2 \sin A \cos A$ |
| Double Angle (Cosine) | Derived | $\cos2A = \cos^2A - \sin^2A = 2\cos^2A - 1 = 1 - 2\sin^2A$ |
| Double Angle (Tangent) | Derived | $\tan2A = \frac{2\tan A}{1 - \tan^2A}$ |

## What's next

Now that you have mastered using addition formulae and deriving double-angle identities, you are ready to apply these skills to solve trigonometric equations, which is the next topic in your Edexcel 4PM1 trigonometry unit. You will also use these identities in later topics such as calculus of trigonometric functions, where simplification using double-angle rules makes integration and differentiation much simpler. Make sure you practice deriving the double-angle identities regularly to build speed for your exam, and always reference the given addition formulae first when attempting any problem involving combined angles.

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