Trigonometric identities in A-Level Maths: how to prove and use them

Ask a Year 13 student which topic in Pure Mathematics feels the most unpredictable and trigonometry will be near the top of the list. Part of the reason is that trig questions rarely announce what they want. A question might look like an equation-solving problem but actually be a disguised identity question, or a proof might only work once you spot that a fraction can be rewritten using a double angle formula. The good news is that the number of identities you genuinely need is small, and the strategies for using them are very consistent from paper to paper.

This guide covers the identities that appear on the Edexcel, AQA, OCR and OCR MEI A-Level Maths specifications, how to approach a 'prove that' question, how to solve trigonometric equations in a given interval, and the specific habits that separate students who score full marks from those who drop two or three marks on every trig question.

The identities you must know by heart

Some identities are printed in the formula booklet and some are not. That distinction matters, because the ones that are not printed tend to be the ones examiners use as the first step of a longer question. If you have to derive them in the exam, you lose time and confidence.

Reciprocal functions: sec, cosec and cot

In Year 13 you meet sec x = 1/cos x, cosec x = 1/sin x and cot x = 1/tan x. Students often treat these as an extra layer of difficulty, but in practice the best strategy is usually to rewrite everything in terms of sin and cos, simplify, and only convert back at the end if the question asks for it. A useful memory aid: the third letter of each reciprocal function tells you what it is the reciprocal of — se(c) goes with cos, co(s)ec goes with sin, co(t) goes with tan.

You should also be able to sketch y = sec x, y = cosec x and y = cot x, including their asymptotes. Sketching questions are relatively quick marks, and the graphs also help you check whether an equation like sec x = 0.5 has any solutions at all. It does not, because sec x is never between −1 and 1.

How to prove a trigonometric identity

Proof questions are where marks are most easily lost through presentation rather than mathematics. An identity proof is not an equation to be solved. You must not move terms from one side to the other as though they were equal already, because that assumes what you are trying to show.

1. Start with the more complicated side. It is far easier to simplify something messy than to make something simple look messy.
2. Rewrite reciprocal functions and tan in terms of sin and cos if you are unsure how to proceed.
3. Look for fractions that can be combined over a common denominator — this is the single most common first step.
4. Look for sin²x + cos²x hiding inside an expression, or a form of cos 2x or sin 2x.
5. Factorise wherever you can. Differences of two squares appear constantly, for example 1 − sin²x or sec²x − 1.
6. Write one step per line, keep an equals sign at the start of each line, and finish with the exact expression on the other side followed by a clear concluding statement.

Here is a typical example. Prove that (1 − cos 2x) / sin 2x ≡ tan x. Start on the left. Replace 1 − cos 2x with 2sin²x, using the form 1 − 2sin²x rearranged. Replace sin 2x with 2 sin x cos x. The expression becomes 2sin²x / (2 sin x cos x). Cancel 2 sin x to leave sin x / cos x, which is tan x. Four lines, each justified, and the proof is complete. The key decision was choosing the version of cos 2x that cancels the 1 — that is exactly the kind of choice the examiner is testing.

Solving trigonometric equations in an interval

Equation-solving questions usually come in two parts: use an identity to rewrite the equation, then solve it in a given interval. Students rarely struggle with the first value; they struggle with finding every value in the interval and with the transformations that change the interval.

Step 1: Reduce to a single trig function

If an equation contains both sin²x and cos x, replace sin²x with 1 − cos²x to get a quadratic in cos x. If it contains sec²x and tan x, use 1 + tan²x = sec²x. If it contains cos 2x and sin x, use cos 2x = 1 − 2sin²x. The aim is always the same: one function, then factorise or use the quadratic formula.

Step 2: Adjust the interval for multiple angles

If you are solving sin 2x = 0.5 for 0 ≤ x < 360°, let θ = 2x and solve for 0 ≤ θ < 720°. Find all values of θ in that larger interval, then halve them. Students who forget this step find only half the solutions and lose most of the marks. The same principle applies to shifts: for cos(x − 30°), shift the interval by 30° before solving.

Step 3: Use the graph or the CAST diagram to find every solution

Your calculator only gives the principal value. Sketch the graph of the relevant function across the interval, or use the CAST diagram, to locate all the other solutions. A quick sketch takes twenty seconds and is far more reliable than trying to remember rules such as '180 minus' for sine and '360 minus' for cosine.

Step 4: Never divide by a trig function you could have factorised

If you have sin x cos x = sin x, do not divide by sin x. Bring everything to one side, factorise as sin x (cos x − 1) = 0, and solve both factors. Dividing loses the solutions where sin x = 0, and examiners design questions specifically to catch this.

Radians, exact values and the R-formula

At A-Level you are expected to work in radians as fluently as in degrees, and calculus with trig functions only works in radians. Make sure your calculator is in the correct mode before every question — it is one of the most common causes of a whole question going wrong.

You should also know the exact values of sin, cos and tan at 0, π/6, π/4, π/3 and π/2 without a calculator. These appear in non-calculator style steps, in exact-answer questions, and as part of compound angle simplifications.

The harmonic form, often called the R-formula, rewrites a sin x + b cos x as R sin(x + α) or R cos(x − α). Expand the target form using the compound angle formula, compare coefficients to get R cos α and R sin α, then R = √(a² + b²) and tan α equals the appropriate ratio. This form is then used to find maximum and minimum values or to solve equations, and those follow-up parts are frequently worth more marks than the rewriting itself.

The mistakes that cost the most marks

• Writing sin²x as sin x² or treating sin(A + B) as sin A + sin B — the latter is never true in general.
• Losing solutions by dividing through by a trig function instead of factorising.
• Forgetting to adjust the interval for multiple or shifted angles.
• Leaving answers in degrees when the interval was given in radians, or vice versa.
• Writing an identity proof that manipulates both sides at once.
• Rounding mid-calculation and getting an answer that falls just outside the required accuracy.
• Ignoring 'hence' — if the question says hence, the previous part is the intended route, and using another method may not earn the marks.

A revision plan for trig identities

Trigonometry rewards short, frequent practice more than long sessions. A sensible two-week plan for a Year 13 student looks like this.

1. Days 1–2: write every identity in the table above from memory, then derive 1 + tan²x = sec²x and the three forms of cos 2x from scratch.
2. Days 3–5: ten identity proofs, one per sitting, written up to full exam standard.
3. Days 6–8: equation solving with multiple angles and reciprocal functions, focusing on finding every solution.
4. Days 9–10: harmonic form questions, including maximum and minimum values.
5. Days 11–14: mixed past paper trig questions under timed conditions, marked strictly against the mark scheme.

For targeted question sets, MathVault has free topic practice you can use for the early days, and MathVault Premium provides worked solutions so you can see exactly where a proof went wrong rather than only knowing that it did.

When trig still is not clicking

If trigonometry keeps falling apart despite practice, the cause is very often underneath it: weak algebraic fractions, unreliable factorising, or a shaky understanding of the unit circle from Year 12. Extra trig questions will not fix that. Diagnosing the real gap and rebuilding it is exactly the kind of work I do in one-to-one lessons — I am a QTS fully qualified teacher and Deputy Head of Maths with over 2,000 hours of tutoring experience, and I teach A-Level students both online and in London.

The short version

1. Memorise the small set of core identities, especially all three forms of cos 2x.
2. In proofs, start with the complicated side and never manipulate both sides.
3. Reduce equations to a single trig function, then factorise rather than divide.
4. Adjust the interval for multiple and shifted angles before solving.
5. Use a sketch to find every solution, and check your calculator mode every time.