Trig identities for A-Level Maths: the full list, with proofs

For A-Level Maths, you need to know these trig identities by heart: sin²x + cos²x ≡ 1, tan x ≡ sin x / cos x, 1 + tan²x ≡ sec²x, 1 + cot²x ≡ cosec²x and the double angle formulae for sin 2A, cos 2A and tan 2A. AQA, Pearson Edexcel and OCR don't print any of these in the formulae they give you in the exam. All three do print the compound angle formulae, sin(A ± B), cos(A ± B) and tan(A ± B). You can build almost everything else from those, including the R sin(x + α) form.

Which trig identities do you need for A-Level Maths?

The Department for Education's subject content for A-Level Maths lists the formulae every exam board must expect you to know without being given them. For trigonometry, that list is:

• the sine rule, the cosine rule and the area of a triangle, ½ab sin C
• sin²A + cos²A ≡ 1
• sec²A ≡ 1 + tan²A and cosec²A ≡ 1 + cot²A
• sin 2A ≡ 2 sin A cos A
• cos 2A ≡ cos²A − sin²A
• tan 2A ≡ 2 tan A / (1 − tan²A)

You also need tan x ≡ sin x / cos x and the definitions sec x ≡ 1/cos x, cosec x ≡ 1/sin x and cot x ≡ 1/tan x. Some lists online talk about "the 8 identities" or "the 12 identities". The number just depends on how they're grouped. For your exam, the DfE list above is the one to learn.

At AS level, you use the sine rule, the cosine rule, sin²x + cos²x ≡ 1 and tan x ≡ sin x / cos x, with angles in degrees. The sec, cosec and cot identities, the double angle formulae and radians are only in the full A-Level.

Which trig identities are in the A-Level formula booklet?

AQA and Pearson Edexcel give you a formula booklet in every A-Level Maths exam. OCR doesn't use a separate booklet. It prints the formulae on pages 2 and 3 of each question paper. Here's which identities you can look up and which ones you need to learn.

Source: DfE subject content for A-Level Maths, appendix B, AQA formulae booklet (v2.0), Pearson Edexcel formulae book and the OCR specification for A Level Mathematics A. Last checked: 10 November 2026.

Every board also prints the small angle approximations, and Pearson Edexcel's booklet adds the sum-to-product formulae, such as sin A + sin B. Check your own board's formulae before the exam so you know exactly what's printed and where.

How to prove the double angle formulae from compound angles

The double angle formulae aren't printed, so it helps to know you can rebuild them in under a minute.

1. Start with sin(A + B) ≡ sin A cos B + cos A sin B. Put B = A to get sin 2A ≡ 2 sin A cos A.
2. Start with cos(A + B) ≡ cos A cos B − sin A sin B. Put B = A to get cos 2A ≡ cos²A − sin²A.
3. Replace sin²A with 1 − cos²A to get cos 2A ≡ 2cos²A − 1.
4. Or replace cos²A with 1 − sin²A to get cos 2A ≡ 1 − 2sin²A.
5. Put B = A in tan(A + B) ≡ (tan A + tan B) / (1 − tan A tan B) to get tan 2A ≡ 2 tan A / (1 − tan²A).

The three forms of cos 2A come up the most. Choose the one that matches the rest of the question. If everything else is in sin, use 1 − 2sin²A. If it's in cos, use 2cos²A − 1.

How to use sec, cosec and cot identities

The sec² identity comes straight from sin²x + cos²x ≡ 1. Divide every term by cos²x and you get tan²x + 1 ≡ sec²x. Divide by sin²x instead and you get 1 + cot²x ≡ cosec²x.

These identities are most useful when an equation mixes two trig functions. Here's an example.

Question (5 marks)

Solve 2 tan²θ + sec θ = 1 for 0° ≤ θ < 360°. Give your answers to 2 decimal places.

Answer

1. Replace tan²θ with sec²θ − 1 to get 2sec²θ − 2 + sec θ = 1.
2. Rearrange into a quadratic: 2sec²θ + sec θ − 3 = 0.
3. Factorise: (2 sec θ + 3)(sec θ − 1) = 0, so sec θ = 1 or sec θ = −3/2.
4. Change to cos: cos θ = 1 or cos θ = −2/3.
5. Solve: θ = 0°, 131.81° or 228.19°, to 2 decimal places.

The value 228.19° comes from 360° − 131.81°. Remember that cos is negative in the second and third quadrants.

How to write a sin x + b cos x in R sin(x + α) form

An expression like 3 sin x + 4 cos x can be written as a single sine function. This makes it easier to solve equations and find maximum values.

Question (3 marks)

Write 3 sin x + 4 cos x in the form R sin(x + α), where R > 0 and 0° < α < 90°. Give the value of α to 2 decimal places.

Answer

1. Expand: R sin(x + α) ≡ R sin x cos α + R cos x sin α.
2. Compare coefficients: R cos α = 3 and R sin α = 4.
3. Square both equations: R²cos²α = 9 and R²sin²α = 16.
4. Add them: R²cos²α + R²sin²α = 9 + 16, so R²(cos²α + sin²α) = 25.
5. Use cos²α + sin²α ≡ 1 to get R² = 25, so R = 5.
6. Divide R sin α = 4 by R cos α = 3: (R sin α) / (R cos α) = 4/3. The R cancels, and sin α / cos α ≡ tan α, so tan α = 4/3.
7. Find α: α = tan⁻¹(4/3) = 53.13°.
8. So 3 sin x + 4 cos x ≡ 5 sin(x + 53.13°).

The maximum value is 5, when x + 53.13° = 90°, so x = 36.87°.

Question (4 marks)

Solve 3 sin x + 4 cos x = 2 for 0° ≤ x < 360°. Give your answers to 2 decimal places.

Answer

1. Use the R form: 5 sin(x + 53.13°) = 2, so sin(x + 53.13°) = 0.4.
2. Change the range: since x goes from 0° to 360°, x + 53.13° goes from 53.13° to 413.13°.
3. Find the first value: sin⁻¹(0.4) = 23.58°, which falls outside that range.
4. Find the values in the range: 180° − 23.58° = 156.42° and 360° + 23.58° = 383.58°.
5. Subtract 53.13° from each: 156.42° − 53.13° = 103.29° and 383.58° − 53.13° = 330.45°.
6. So x = 103.29° or x = 330.45°, to 2 decimal places.

Students often forget to change the range. Write the new range down before you start solving.

How to answer a trig "prove that" question

Here's a typical question.

Question (4 marks)

Prove that (1 − cos 2x) / sin 2x ≡ tan x.

Answer

1. Start with the more complicated side, which is the left. There's a 1 on top, so use the form of cos 2x that will cancel it: cos 2x ≡ 1 − 2sin²x. On the bottom, use sin 2x ≡ 2 sin x cos x.
2. LHS ≡ (1 − (1 − 2sin²x)) / (2 sin x cos x)
3. ≡ 2sin²x / (2 sin x cos x)
4. ≡ sin x / cos x
5. ≡ tan x ≡ RHS

A few habits that help with every proof:

• Work on one side only. Don't treat it like an equation and do the same thing to both sides.
• Write out each line in full, and say which identity you've used.
• If you're stuck, rewrite everything in terms of sin and cos.
• Finish by stating that the left side equals the right side.

If you want to practise questions like this, Medly AI covers A-Level Maths for AQA, Edexcel and OCR. The free plan includes 5 marked questions, 20 AI tutor messages and 1 uploaded paper a day, with no card needed and no end date.

Common trig identity mistakes in A-Level exams

Pearson's examiner report for A-Level Maths Paper 1 in summer 2019 picked out three common trig mistakes.

1. Dividing by sin x or cos x. The examiners noted that most students divided through by sin or cos, which meant they lost the solutions where it equals zero. Factorise instead. For example, sin x(2 cos x − 1) = 0 gives sin x = 0 as well as cos x = ½.
2. Stopping at the first answer. Many students found the acute angle from tan x = 4 and didn't look for the second solution. Sketch the graph or use a CAST diagram to find every solution in the range.
3. Mixing up degrees and radians. Using degrees when the question needed radians was a common error. If the range is written with π, set your calculator to radians.

Practising exam questions and marking them against the mark scheme is a good way to stop making these mistakes. You'll find past papers in our past papers directory.

Frequently asked questions

Are the double angle formulae in the A-Level formula booklet?

No. AQA, Pearson Edexcel and OCR don't print them, so you need to learn them. If you forget, you can work them out by putting B = A in the compound angle formulae, which are printed.

Do you need to learn trig identities for A-Level Maths?

Yes. You need to know sin²x + cos²x ≡ 1, tan x ≡ sin x / cos x, the sec² and cosec² identities, the double angle formulae and the sine and cosine rules. The compound angle formulae are given to you in the exam.

What is the difference between AS and A-Level trig identities?

AS Maths uses sin²x + cos²x ≡ 1 and tan x ≡ sin x / cos x, with angles in degrees only, while the full A-Level adds radians, sec, cosec and cot, the compound and double angle formulae and the R form.

How do you know which trig identity to use?

Look at what's in the question. If tan² and sec appear together, use the sec² identity. If you see 2x and x, use a double angle formula, and if you see a sin x + b cos x, use the R form. If none of these fit, rewrite everything in sin and cos.

Are trig identities on the non-calculator paper?

There isn't a non-calculator paper in A-Level Maths with AQA, Pearson Edexcel or OCR. You can use a calculator on every paper, but proofs and "show that" questions still need every step written out to get the marks.

Try Medly for free