How to Answer "Show That" Questions in Cambridge AS & A Level Mathematics (9709)

A “show that” question gives you the result you need to reach. Your job is to explain, through valid mathematical working, how the information in the question leads to it. Start from what you know, show the important transformations, and finish in the requested form. Copying the printed answer or producing a matching calculator value does not demonstrate the reasoning. These questions can feel frustrating because you already know what the last line should say. The difficulty is making the route to that line convincing. This guide explains how to build that route, how much working to include, and how to handle exact values and follow-up questions. The written practice examples below are original NeuraGeek questions. The screenshots show how related Cambridge past-paper questions can be explored inside NeuraGeek.

What “show that” asks you to do

Cambridge’s 2026–2027 Mathematics 9709 syllabus defines “Show (that)” as “provide structured evidence that leads to a given result”. That means a successful response needs a connected argument. In algebra, that might be a sequence of equivalent expressions; in mechanics, equations based on the forces acting; in probability, a calculation built from the conditions of the distribution. Before writing, identify both your starting information and your target. Is the question asking for an equation, an exact value, an identity, or a relationship between quantities? Notice any restrictions, such as a positive variable or an interval for an angle. These details help you select a method and avoid introducing a step that is not valid for the question.

Start from the information you are given

A useful default is to work forward from the question. You can inspect the target while planning: a squared expression may suggest completing the square, while a trigonometric fraction may suggest an identity. But your finished solution should explain why the target follows, without assuming the very result you have been asked to establish. NeuraGeek practice question: A circle has centre (2, −3) and passes through the origin. Show that its equation is x² + y² − 4x + 6y = 0. An incomplete response would simply write the target equation and say that it is the required circle. A clearer solution calculates the radius from the supplied centre and point, then uses the standard equation of a circle:

r² = (0 − 2)² + (0 + 3)² = 13

(x − 2)² + (y + 3)² = 13

x² − 4x + 4 + y² + 6y + 9 = 13

x² + y² − 4x + 6y = 0

The working establishes the radius, substitutes into a relevant formula, and expands to reach the requested form. Each line has a purpose. There is no need to add a separate sentence explaining every numerical addition; the important point is that the mathematical connection is visible. Working backwards is not automatically invalid mathematics. Reversible transformations can establish equivalence. The danger is assuming the target is true and reaching something familiar without showing that the argument works in the required direction. When you are unsure, use backwards exploration as rough work, then write a clear forward derivation.

Show the transitions that carry the argument

The amount of working depends on what the method involves. A useful self-check is to ask whether a reader can follow each transition without having to reconstruct the main calculation. Expanding a complicated bracket, substituting a new variable, combining fractions, or choosing a physical law may deserve an explicit line. Repeating an unchanged expression usually adds little. NeuraGeek practice question: For angles θ with sin θ ≠ 0, show that (1 − cos² θ)/sin θ = sin θ. A weak response copies both sides with an equals sign between them. A clear response starts with the left-hand expression and applies the identity sin² θ + cos² θ = 1:

(1 − cos² θ)/sin θ

= sin² θ/sin θ

= sin θ, since sin θ ≠ 0.

This short solution is sufficient to explain the mathematical argument: it identifies the substitution and makes the cancellation valid. More lines would not necessarily make it better. By contrast, testing θ = 30° would only check one allowed angle. It would not establish the identity for every angle in its domain. Treat the target as a check on your work, too. If your expansion produces the wrong coefficient, return to the previous line and look for the error. Do not change a sign simply to match the printed result. Check negative signs, brackets, powers and denominators before repeating the entire calculation.

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Keep exact values exact when required

An exact answer retains the full mathematical value, for example √3, π/6 or 2/7. A rounded decimal is an approximation. If the target is exact, keep fractions, surds and other exact quantities through the derivation. This lets the final simplification establish the stated value without relying on two decimals looking close. NeuraGeek practice question: Show that the definite integral of x² from 0 to √3 is √3. Using the antiderivative x³/3 gives:

Antiderivative: x³/3

Integral = (√3)³/3 − 0

= 3√3/3

= √3

The key simplification uses (√3)³ = 3√3. Replacing √3 with 1.732 at the start turns an exact calculation into an approximate one. You could still use a decimal to check the size of your answer privately, but the written derivation should preserve the exact relationship. Not every “show that” target is exact. Some questions give an approximate numerical value. Follow the wording and stated accuracy, and keep unrounded intermediate values when calculating. Cambridge’s syllabus explicitly advises against rounding before the final answer when accuracy marks are at stake.

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Use your calculator to support the working

A calculator can help check arithmetic or alert you to an implausible value. It cannot make an unexplained line into a derivation. The current syllabus expects necessary working and states that unsupported calculator answers receive no marks. Scientific calculators are expected, subject to Cambridge’s restrictions on permitted devices. For a calculation, write the relationship you are using and show how the question’s values enter it. For an integral, show the antiderivative and evaluation where that is your method. For an algebraic result, make the transformations visible. This also makes mistakes easier to locate when you review your attempt afterwards.

Connect the result to the next part

When a following part begins with “hence”, look for how the result you have just established helps you proceed. Perhaps an equation has been rearranged into a quadratic, an identity simplifies a new expression, or a value can now be substituted into a formula. Make that connection explicit in your working rather than treating each subpart as an unrelated problem. “Hence or otherwise” signals that another valid route is available. Even then, check the previous result before starting a longer method. If you could not complete the earlier proof, a printed result can still help you tackle a later part that asks you to use it. Using that supplied result later does not, by itself, complete the missing proof.

Check your answer and practise the missing step

Before moving on, check that you started from valid information, used justified transformations, kept any required exact values, and reached the stated form. Check restrictions before dividing or cancelling, and make sure a calculator output has not replaced an essential explanation. There is no fixed line count to aim for; aim for a complete, readable argument. To practise, open Mathematics 9709 in NeuraGeek and attempt a “show that” question from Past Papers or Topical Papers before viewing the solution. Submit your working through the AI Tutor’s Examiner mode and use the feedback to identify a gap, such as an unexplained substitution or an algebraic error. Rewrite that section, then attempt another question independently. The useful improvement is being able to produce the reasoning yourself the next time.