Trig Identities You Must Know for A-Level (and How to Use Them)

Quick answer: The trig identities you must know for A-Level are the Pythagorean identity sin^2 x + cos^2 x = 1 (plus its cousins 1 + tan^2 x = sec^2 x and 1 + cot^2 x = cosec^2 x), the quotient identity tan x = sin x / cos x, the double-angle formulae, and the addition formulae. You use them to turn a messy trig equation into one function you can actually solve.

When I first hit trig identities in A-Level I thought the whole thing was just memorising a wall of formulae. It isn't. The exam almost never asks you to recite one. It hides the identity inside a "solve this equation" or "prove that" question, and the real skill is spotting which identity unlocks it. Here is the shortlist I actually use, plus how I decide which one to reach for under pressure.

The identities you genuinely have to know by heart

Pythagorean family:

• sin^2 x + cos^2 x = 1
• 1 + tan^2 x = sec^2 x
• 1 + cot^2 x = cosec^2 x

You get the last two for free by dividing the first one by cos^2 x and then by sin^2 x, so if you only remember the top line you can rebuild the other two in the margin.

Quotient identity:

• tan x = sin x / cos x (and cot x = cos x / sin x)

Double-angle formulae:

• sin 2x = 2 sin x cos x
• cos 2x = cos^2 x - sin^2 x = 2 cos^2 x - 1 = 1 - 2 sin^2 x
• tan 2x = 2 tan x / (1 - tan^2 x)

Addition formulae:

• sin(A + B) = sin A cos B + cos A sin B
• cos(A + B) = cos A cos B - sin A sin B
• tan(A + B) = (tan A + tan B) / (1 - tan A tan B)

Flip the signs for the minus versions. Here is a genuinely useful exam-board detail: on Edexcel, AQA and OCR A-Level, and on Cambridge International A-Level, the addition and double-angle formulae are printed in the formula booklet, but the Pythagorean and quotient identities are not. So those two families have to live in your head. AP Precalculus and AP Calculus lean on the same set, so this list travels well.

How to decide which identity to use

This is the part nobody teaches properly. Match the shape of the question to the tool:

• You see a "2x" sitting next to a plain "x"? Use a double-angle formula so everything is in terms of x.
• You see sin^2 or cos^2 and want a single function? Reach for sin^2 x + cos^2 x = 1.
• You see sec, cosec or cot? Rewrite using 1 + tan^2 x = sec^2 x (or its cot version).
• You see a sin x + b cos x? Use the R formula, a sin x + b cos x = R sin(x + alpha).
• Proving an identity? Start on the messier side and drive it toward the simpler side. Never work on both sides at once, because the mark scheme wants one continuous chain of equals signs.

For cos 2x specifically, there are three versions. Pick the one that matches the other terms in the equation. If the rest of the equation is in sin, use cos 2x = 1 - 2 sin^2 x so everything becomes sin.

Worked example: solve cos 2x + sin x = 0 for 0 ≤ x ≤ 360°

Step 1. Spot the mismatch. There is a 2x and a plain x, so replace cos 2x. Because the other term is sin x, choose the sin version:

cos 2x = 1 - 2 sin^2 x

Step 2. Substitute:

1 - 2 sin^2 x + sin x = 0

Step 3. Rearrange into a standard quadratic (multiply through by -1 so the squared term is positive):

2 sin^2 x - sin x - 1 = 0

Step 4. Factorise it exactly like 2y^2 - y - 1:

(2 sin x + 1)(sin x - 1) = 0

Step 5. Solve each bracket:

sin x = -1/2 or sin x = 1

Step 6. Read off every solution in the range. sin x = 1 gives x = 90°. sin x = -1/2 is negative, so x is in the third and fourth quadrants: x = 210° and x = 330°.

Final answer: x = 90°, 210°, 330°.

The classic lost mark here is stopping at sin x = -1/2 and only writing 210°. Always sketch the sin curve or use the CAST diagram to sweep the whole range.

Worked example: the R formula

Express 3 sin x + 4 cos x in the form R sin(x + alpha), with R greater than 0 and alpha between 0 and 90°.

Step 1. Expand the target form:

R sin(x + alpha) = R sin x cos alpha + R cos x sin alpha

Step 2. Compare coefficients with 3 sin x + 4 cos x:

R cos alpha = 3 and R sin alpha = 4

Step 3. Find R by using cos^2 + sin^2 = 1, which means R = sqrt(3^2 + 4^2):

R = sqrt(25) = 5

Step 4. Find alpha by dividing the two equations, since sin/cos = tan:

tan alpha = 4/3, so alpha = 53.1° (1 d.p.)

Result: 3 sin x + 4 cos x = 5 sin(x + 53.1°).

The examiner trap is rounding alpha too early and then using it to solve an equation, which drags your final angles off by a degree or two. Keep the unrounded value in your calculator until the very last line.

Common mistakes examiners flag

• Dividing an equation by sin x or cos x. You lose the solutions where that function equals zero. Factorise instead.
• Giving only one answer when the range clearly holds more.
• Choosing the wrong cos 2x version and creating extra algebra.
• Working on both sides of a "prove" question at once. Pick one side and transform it.

Test yourself

1. Solve 2 cos^2 x - 1 = sin x for 0 ≤ x ≤ 360°.
2. Given tan theta = 3/4 and theta is acute, find the exact value of sin 2theta.
3. Prove that (1 - cos 2x) / sin 2x = tan x.

Quick answers:

1. Use cos 2x = 2 cos^2 x - 1, so the left side is cos 2x = sin x. Rewrite cos 2x = 1 - 2 sin^2 x to get 2 sin^2 x + sin x - 1 = 0, then (2 sin x - 1)(sin x + 1) = 0. Solutions: x = 30°, 150°, 270°.
2. sin theta = 3/5, cos theta = 4/5, so sin 2theta = 2 sin theta cos theta = 24/25.
3. Top becomes 1 - (1 - 2 sin^2 x) = 2 sin^2 x. Bottom is 2 sin x cos x. Cancel to sin x / cos x = tan x.

Want each step checked instantly? Paste any of these into the math solver and it walks through the working line by line, so you can see exactly where a solution goes missing.

FAQ

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

On Edexcel, AQA, OCR and Cambridge International A-Level, the addition formulae and double-angle formulae are printed. The Pythagorean identities (sin^2 x + cos^2 x = 1 and friends) and the quotient identity tan x = sin x / cos x are not, so memorise those. If in doubt, check your board's most recent formula sheet before the exam.

How do I know when to use the double-angle formula versus the Pythagorean identity?

Look at the angles. If the equation mixes 2x and x, a double-angle formula makes the angles match. If every angle is already the same but you have a squared term, the Pythagorean identity reduces it to one function. Many questions need both, one after the other.

What is the fastest way to memorise all of them?

Learn sin^2 x + cos^2 x = 1 and the addition formulae properly, then derive the rest. The double-angle formulae are just the addition formulae with B = A, and the sec and cosec identities come from dividing the Pythagorean line. Turning them into flashcards and testing the derivations, not just the results, is what made them stick for me.

Why do I keep losing marks on "solve" questions even when my algebra is right?

Usually the range. Once you have sin x or cos x equal to a value, find every angle in the given interval using a quick curve sketch or the CAST diagram. A timed mock exam is the fastest way to spot whether you are habitually dropping the second or third solution.

In short: memorise the Pythagorean and quotient identities, know how to rebuild the double-angle and addition formulae, and practise matching the shape of a question to the right identity. Nail that, and identity questions turn from guesswork into free marks. Check your working step by step in the math solver whenever a solution feels off.