Addition Formulae

Edexcel International GCSE Further Pure Mathematics· 10G· 15 min read

1. Given Addition Formulae and Basic Application★★☆☆☆⏱ 4 min

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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.

📘 Definition

Addition Formulae (Given)

The six core provided formulae are:

$sin (A + B) = sin A cos B + cos A sin B$
$sin (A - B) = sin A cos B - cos A sin B$
$cos (A + B) = cos A cos B - sin A sin B$
$cos (A - B) = cos A cos B + sin A sin B$
$tan (A + B) = \frac{t a n A + t a n B}{1 - t a n A t a n B}$
$tan (A - B) = \frac{t a n A - t a n B}{1 + t a n A t a n B}$

📐 Worked Example

Calculate the exact value of $sin 7 5^{\circ}$ using the addition formula for $sin (A + B)$ .

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Step 1: Express 75° as the sum of two angles with known exact values: $7 5^{\circ} = 4 5^{\circ} + 3 0^{\circ}$ , so $A = 4 5^{\circ}$ , $B = 3 0^{\circ}$
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Step 2: Substitute into the given $sin (A + B)$ formula:
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$\sin(45^\circ + 30^\circ) = \sin45^\circ \cos30^\circ + \cos45^\circ \sin30^\circ$
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Step 3: Substitute exact values: $sin 4 5^{\circ} = \frac{2}{2}$ , $cos 3 0^{\circ} = \frac{3}{2}$ , $cos 4 5^{\circ} = \frac{2}{2}$ , $sin 3 0^{\circ} = \frac{1}{2}$
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$= \left(\frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2} \times \frac{1}{2}\right) = \frac{\sqrt{6} + \sqrt{2}}{4}$

Exam tip:

Always use exact values from 30/45/60 angles when asked for exact results, never use decimal approximations.

2. Deriving Double-Angle Identities from Addition Formulae★★★☆☆⏱ 4 min

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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.

📘 Definition

Double-Angle Identities (Derived)

You must be able to derive all of these from the given addition formulae:

$sin 2 A = 2 sin A cos A$
$cos 2 A = cos^{2} A - sin^{2} A = 2 cos^{2} A - 1 = 1 - 2 sin^{2} A$
$tan 2 A = \frac{2 t a n A}{1 - t a n ^{2} A}$

📐 Worked Example

Derive the identity for $cos 2 A$ in terms of $cos^{2} A$ only, starting from the $cos (A + B)$ addition formula.

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

3. Expanding and Simplifying Trigonometric Expressions★★★☆☆⏱ 4 min

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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 $cos (θ + 3 0^{\circ}) + sin (θ - 6 0^{\circ})$ , leaving your answer in its simplest form.

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

Exam tip:

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

4. Verifying Trigonometric Identities★★★★☆⏱ 3 min

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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 $sin (A + B) + sin (A - B) = 2 sin A cos B$ .

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

5. Common Pitfalls

Wrong move:

Mixing up the sign in the $cos (A \pm B)$ formula

Why:

The sign in the cos expansion is the opposite of the operation between the angles, so students often write + instead of -

Correct move:

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 move:

Forgetting to derive double-angle identities if asked explicitly

Why:

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 move:

Always set $B = A$ in the relevant addition formula and show every step of the derivation when requested.

Wrong move:

Using $tan (A \pm B)$ formula when $A$ or $B$ is an odd multiple of $9 0^{\circ}$

Why:

Tangent is undefined at odd multiples of $9 0^{\circ}$ , so the formula will produce an invalid result

Correct move:

Use sine and cosine addition formulae instead for angles where $tan$ is undefined.

Wrong move:

Incorrectly carrying signs when expanding $sin (A - B)$ or $cos (A - B)$

Why:

Students often forget to apply the negative sign to all terms in the second half of the expansion

Correct move:

Write the angle as $A + (- B)$ and use the standard addition formula to preserve sign consistency if you struggle with the subtraction form.

Wrong move:

Using degree mode on your calculator when angles are given in radians, or vice versa

Why:

This will produce incorrect numerical values and invalidate your working

Correct move:

Always check the angle unit given in the question, and adjust your calculator mode accordingly if you are using a calculator for calculation.

6. Quick Reference 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	$tan (A \pm B) = \frac{t a n A \pm t a n B}{1 \mp t a n A t a n B}$
Double Angle (Sine)	Derived	$sin 2 A = 2 sin A cos A$
Double Angle (Cosine)	Derived	$cos 2 A = cos^{2} A - sin^{2} A = 2 cos^{2} A - 1 = 1 - 2 sin^{2} A$
Double Angle (Tangent)	Derived	$tan 2 A = \frac{2 t a n A}{1 - t a n ^{2} A}$

7. Frequently Asked

Do I need to memorise the addition formulae for 4PM1?

No, the sin(A±B), cos(A±B) and tan(A±B) formulae are provided on your exam formula sheet. You only need to memorise how to derive double-angle identities from them.

Can I use double-angle formulae directly in answers?

Yes, you may use them for simplification and calculation, but if a question explicitly asks you to derive them from addition formulae, you must show all derivation steps to get full marks.

Going deeper

formula sheetEdexcel 4PM1 Official Specification & Formula SheetSee page 32 for given trigonometric formulae

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.