The 12 pure identities every UK 9MA0 student must have on lockdown

Every May, examiners write essentially the same identity questions. The command word changes, the surrounding context changes, but the underlying identity is one of twelve. Memorise these twelve — and know when to reach for each one — and you unlock 15-20 marks of easy points across Papers 1 and 2.

The twelve you must know cold

Trigonometric (5)

• Pythagorean identity: $\sin^2\theta + \cos^2\theta \equiv 1$
• Tangent form: $\tan\theta \equiv \dfrac{\sin\theta}{\cos\theta}$
• Secant Pythagorean: $1 + \tan^2\theta \equiv \sec^2\theta$
• Cosecant Pythagorean: $1 + \cot^2\theta \equiv \csc^2\theta$
• Double-angle for cosine: $\cos 2\theta \equiv 1 - 2\sin^2\theta \equiv 2\cos^2\theta - 1$

Logarithm and exponent (4)

• Log product: $\log_a(xy) \equiv \log_a x + \log_a y$
• Log quotient: $\log_a(x/y) \equiv \log_a x - \log_a y$
• Log power: $\log_a(x^n) \equiv n\log_a x$
• Change of base: $\log_a x \equiv \dfrac{\log_b x}{\log_b a}$

Algebra (3)

• Difference of squares: $a^2 - b^2 \equiv (a-b)(a+b)$
• Sum of cubes: $a^3 + b^3 \equiv (a+b)(a^2 - ab + b^2)$
• Binomial-style: $(a+b)^2 \equiv a^2 + 2ab + b^2$

The three that go wrong most often

1. Sign flip on double-angle cosine. Students remember $\cos 2\theta \equiv 2\cos^2\theta - 1$ but write $1 - 2\cos^2\theta$ under pressure. Whenever you use this identity, circle the "1" and write the sign next to it before touching anything else.

2. Missing modulus on logs. When you divide through by a log, some students forget that $\log_a x$ is only defined for $x > 0$. In a "solve" question, always write the domain restriction next to the manipulation. Examiners award a mark for stating it.

3. Change-of-base direction. The formula is $\log_a x = \dfrac{\log_b x}{\log_b a}$ — students routinely invert it. The base you're changing FROM stays on top; the base you're changing TO goes underneath. Say it out loud twice before you write.

How to drill these

Set a five-minute timer. Fold a piece of paper into twelve boxes, write only the LEFT-hand side of each identity, and hand-write the RIGHT-hand side without looking. Do this three days in a row and you'll have them for life.

Where they show up in the paper

Roughly a quarter of Paper 1 questions rely on one of these twelve identities somewhere in the working. In Q6-8 (worth 6-9 marks each), the identity is often the first step of the method. Get it wrong and every subsequent mark is at risk.

Your action plan

Print the twelve identities. Stick them above your desk. For every past paper you attempt in the next four weeks, notice which identities you actually reach for. Add a tally next to each. By the end of the four weeks, you'll know exactly which of the twelve are your gaps — and that's what you drill in the final fortnight.