The trig identities worth knowing cold

Every SPM Add Math paper is issued with a formula list, and nine trigonometric identities sit on it: the Pythagorean identity and its two variants, the three addition formulae, and the three double-angle formulae. That single fact changes how you should spend your revision time.

There is no mark for reciting $\sin(A+B)=\sin A\cos B+\cos A\sin B$ from memory when it is printed in front of you, the marks sit entirely in choosing the right identity for the question and applying it correctly under pressure.

"Prove that" questions unsettle students who are used to solving for a value, because there is no single number to chase, the goal is to show two expressions are equal for every valid angle. The method is more disciplined than it looks once you have a fixed starting point.

1. 1

   Pick the messier side

   Start working from whichever side of the identity has more terms, a fraction, or a mix of trig ratios. It is almost always easier to simplify a complicated expression down than to build a simple one up.

2. 2

   Convert everything to sin and cos if stuck

   $\tan$, $\sec$ and $\cot$ can each be rewritten in terms of $\sin$ and $\cos$. When a proof stalls, rewriting every ratio this way and simplifying the resulting fraction usually unsticks it.

3. 3

   Reach for the Pythagorean identity to swap terms

   $\sin^{2}A+\cos^{2}A=1$ lets you replace $\sin^{2}A$ with $1-\cos^{2}A$, or the reverse, whenever a substitution would simplify the working, this single move unlocks a large share of proof questions.

4. 4

   Finish by matching the other side exactly

   The proof is only complete once your working reaches the exact form of the other side, not just something equivalent to it. State that clearly as the final line.

$\cos 2A$ is given in three equivalent forms, and knowing which one to reach for is a skill in its own right.

$$\cos 2A = \cos^{2}A - \sin^{2}A = 2\cos^{2}A - 1 = 1 - 2\sin^{2}A$$

• Use $\cos^{2}A-\sin^{2}A$ when the question already has both squared ratios present.
• Use $2\cos^{2}A-1$ when you want to end up with cosine only, common in equations solved for $\cos A$.
• Use $1-2\sin^{2}A$ when the question is phrased in terms of sine, or when it needs rearranging into $\sin^{2}A=\dfrac{1-\cos 2A}{2}$.

That last rearrangement is the "backwards" use worth practising deliberately: reading $\cos 2A=1-2\sin^{2}A$ as a way to express $\sin^{2}A$ in terms of $\cos 2A$, rather than the other direction. It resurfaces later whenever a squared trig term needs to be broken down into a single angle instead of a doubled one, including in some integration questions in Form 5.

The identities themselves are given, so lost marks on this topic rarely come from a wrong formula. They come from a small set of habits worth checking for on purpose.

• Losing a sign when expanding $\cos(A-B)=\cos A\cos B+\sin A\sin B$, the sign flips relative to $\cos(A+B)$, and it is easy to copy the wrong one under time pressure.
• Substituting $\sin^{2}A=1-\cos^{2}A$ correctly but then forgetting to simplify the resulting difference of two squares.
• Stopping one line early, reaching an expression that is equivalent to, but not written identically as, the target side.

Trigonometric identities are not confined to standalone proof questions. They quietly do the work inside several other question types across both papers.

• Solving trigonometric equations, an equation mixing $\sin\theta$ and $\cos 2\theta$ usually needs a double-angle substitution before it can be solved with a single ratio.
• Sketching and reading trig graphs, recognising that $\cos 2\theta$ completes a full cycle twice as fast as $\cos\theta$ comes directly from the identity, not from plotting points.
• Simplifying expressions before differentiating or integrating them in Form 5 calculus, where a product like $\sin\theta\cos\theta$ is often rewritten as $\tfrac{1}{2}\sin 2\theta$ first.

Recognising an identity as the hidden first step in a longer question, rather than only when the words "prove that" appear, is often what separates a question that looks unfamiliar from one that is actually routine.

Trig identities reward pattern recognition more than raw memory, and pattern recognition is exactly the kind of thing that develops faster with someone watching your working in real time. A teacher can see, within one or two proofs, whether you default to converting everything to sine and cosine, whether you spot a Pythagorean substitution quickly, and where your specific hesitation is, then build practice around that gap rather than a generic worksheet.

Our teachers are experienced; lessons run online and are taught in English, while SPM papers themselves are set bilingually in Bahasa Melayu and English.

If trigonometric identities, or trigonometric functions more broadly, are a recurring sticking point, a one-hour paid trial class at the teacher's own rate (from RM50 per hour, depending on experience) is a reasonable way to see how that works before committing further. We will not promise a particular grade, but we can help the nine given identities stop feeling like a list to search through and start feeling like tools you reach for on instinct.