How to Answer "Show That" and Proof-Style Questions in IGCSE Additional Mathematics

“Show that” questions in Cambridge IGCSE Additional Mathematics (0606) are not asking you to discover the final result; the result is already given. They are asking you to justify it. Under the current 2025–2027 syllabus, mathematical communication and structured reasoning are explicit parts of the course, and recent papers use “show that”, “hence” and proof-style tasks across coordinate geometry, trigonometric identities, functions, vectors, logarithms, calculus and counting. The safest approach is to treat the printed result as the destination. Start from information you are genuinely allowed to use, apply valid transformations, keep exact forms when the question requires them and make the chain of reasoning easy to inspect.

The current papers genuinely use this skill

The title is not built around one isolated question type. Current 2025 papers contain repeated examples: showing that a line is tangent to a circle, converting a trigonometric expression into a stated form, proving a relationship obtained from a straight-line graph, showing that a curve has exactly one stationary point, deriving a quadratic expression for the distance between two moving particles and then using it to show that they do not collide, and showing that a counting problem leads to a particular quadratic equation. The topic varies, but the assessment skill is stable: the examiner needs to see why the stated result follows.

Recent 2025 papers show that proof-style reasoning appears across several parts of the 0606 syllabus, not one isolated topic.

A given answer is not a free mark

The most important mindset is simple: do not treat the result printed in the question as an assumption you may immediately use. Cambridge’s March 2025 examiner report says that when a question uses the phrase “Show that”, candidates need a clear, correct and complete method without error or contradiction. June 2025 examiner feedback makes the same point in another form: when candidates were asked to show a derivative in a particular form, they needed enough detail in the final algebraic manipulation to justify that form. Reaching the printed expression without showing how you got there can therefore lose the very marks the question is testing.

Start from known information whenever possible

A forward derivation is usually the clearest exam route. Begin with the equation, identity, coordinates, function or theorem the question has actually given you. Then move one valid step at a time until the target appears. This avoids circular reasoning, where you effectively assume the statement is true and then rearrange it back into something already known. Working backwards is not automatically mathematically invalid, but in an exam it is risky unless every transformation is reversible and you still present a complete argument. For most IGCSE “show that” questions, starting from the known side is safer and easier to mark.

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Make transformations inspectable

Do not jump from the first line to the final target with several algebraic steps hidden in your head or calculator. The 2025 examiner reports repeatedly reward clear, logical progression, and one report specifically notes that candidates needed to show the factorising or division step that converted a derivative into the required form. That does not mean writing every trivial arithmetic operation. It means showing the step that changes the mathematical structure: factorising, completing the square, applying an identity, substituting a known relation, differentiating, integrating, comparing coefficients or forming a discriminant.

Preserve exactness when the target is exact

If the question asks you to show an exact result, do not introduce decimals unless there is a clear reason. Fractions, surds, powers, logarithms and exact trigonometric values often need to survive until the target has been reached. A decimal approximation can hide the structure you were meant to demonstrate and may make it impossible to establish the printed expression exactly. This is especially important on the current non-calculator paper, where exact manipulation is part of the design, but the same principle applies on calculator questions when the required result is symbolic.

Use 'hence' as a bridge, not as a new start

Recent 0606 papers often pair a derivation with a follow-up “hence” part. In one 2025 vector question, candidates first derive a quadratic expression for the squared distance between two particles and then use that result to show that the particles never collide. The word “hence” is a signal that the previous result is meant to do work for you. Do not throw it away and solve the second part from scratch unless there is a strong reason. Ask what the earlier expression now lets you conclude more efficiently.

Proof-style questions are broader than the words 'show that'

The same reasoning habits appear in questions that ask you to prove, justify, establish that only one solution exists, determine the nature of a stationary point or demonstrate that a geometric condition holds. For example, showing that a line is tangent may require you to prove a discriminant is zero or establish perpendicularity at the point of contact. Showing that an equation has one real root may require a discriminant argument or a monotonicity argument. The command changes, but the structure remains: identify what would logically establish the claim, then build that argument visibly.

A worked original example

Consider the original practice statement: given x + 1/x = 3, show that x² + 1/x² = 7. A weak response starts from x² + 1/x² = 7, adds 2 and concludes that (x + 1/x)² = 9. That only checks that the target is consistent with the given information; it does not clearly derive the target from what was known. A stronger response squares the given equation first: (x + 1/x)² = 9, expands to x² + 2 + 1/x² = 9, and then subtracts 2 to reach x² + 1/x² = 7. The difference is direction and justification, not length.

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Do not force a fixed number of lines

There is no universal rule that a “show that” answer needs three lines, five lines or a paragraph of explanation. A two-mark completing-the-square task may need only a compact algebraic sequence, while a five- or six-mark calculus or geometry proof may need several linked stages. The question and mark scheme decide the amount of evidence required. Your aim is not to hit a line count. It is to make every important logical move visible enough that an examiner can follow it without guessing what happened between lines.

Common ways students lose marks

The recurring problems are surprisingly consistent. Students start from the target and create a circular argument; skip the identity or theorem that makes the next step valid; change both sides of an equation inconsistently; lose exact form too early; use a previous result incorrectly in a “hence” part; or reach the target but omit the final factorisation, simplification or sign argument that actually proves it. Examiner reports also note that work becomes difficult to follow when steps are poorly set out. If your argument is correct but hard to trace, you are making the examiner do work that should have been done on the page.

A quick exam checklist

Before leaving a proof-style question, check four things. Did I start from information I was entitled to use? Does each line follow from the previous one by a valid operation? Have I preserved exactness where the question needs an exact result? And have I explicitly reached the target rather than stopping one algebraic step short? For “hence” questions, add a fifth check: did I actually use the previous result? These questions are often less about difficult arithmetic than about disciplined communication.

Frequently asked questions

Can I work backwards in a “show that” question? Sometimes, if every step is logically reversible and your final argument is still complete, but a forward derivation is usually safer. Do I need to copy the target at the end? It is good practice to make the final matching form explicit. Can I use my calculator to verify the result? You can use a calculator as a check on calculator papers, but numerical verification does not replace the required derivation. Does “hence” mean I must use the previous part? It strongly signals that the previous result is intended to be used. Are all proof-style questions marked the same way? No. The exact method and marks depend on the question and mark scheme.

The practical takeaway

Treat “show that” as a reasoning task, not a result task. Begin from known information, use valid identities and transformations, preserve exact forms, make the key algebra visible and let the printed target be the final destination. Current 0606 papers use this skill across many topics, and current examiner reports emphasise clear, complete mathematical communication. If the examiner can see why each step follows and why the last line matches the stated result, you are doing what the question is designed to test.