How to Answer Proof Questions in Add Math

Proof questions unsettle students because there’s no “answer” to compute, you’re constructing an argument. But 0606 proofs, mostly trigonometric identities, follow a fixed, learnable discipline. The marks are for a clear logical chain, and laying that chain out correctly is a technique, not a talent.

Start from a definite point and move one direction

The cardinal rule for an identity proof: work down one side until it becomes the other. Start from the more complicated side (there’s more to simplify) and manipulate it step by step toward the target. Working both sides at once, “meeting in the middle”, risks the logical error of assuming the very thing you’re proving, and examiners penalise it.

A typical identity proof runs: combine fractions over a common denominator, expand, then apply $\sin^2\theta + \cos^2\theta = 1$ to collapse the expression to the required form. Each line follows logically from the last, no leaps.

Name your tools

Most 0606 proofs run on a small toolkit:

• The Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ (and its rearrangements).
• The quotient identity $\tan\theta = \frac{\sin\theta}{\cos\theta}$.
• Common-denominator algebra and factorising.

When a proof stalls, the fix is almost always one of these, usually converting everything to sines and cosines and applying the Pythagorean identity. Knowing the toolkit turns “where do I start?” into a short checklist.

Show every step, and don’t assume the conclusion

As with show that questions, every line must be shown, a step done in your head is a mark lost. And the logic must flow toward the conclusion, never using the result to justify an earlier line. Keep the direction clean and the proof is watertight.

End explicitly

Finish by stating that you’ve reached the required result, ”$= \frac{2}{\sin\theta}$, as required”, or a small QED mark. It signals to the examiner that the chain is complete, and confirms to yourself that you landed on the target rather than something close to it.

Proof questions become reliable marks once the discipline is habitual, and habit comes from doing several with feedback. A specialist can turn “I never know how to start” into a routine within a session or two, which is exactly what the trial class begins. See the trig identities notes and the technique guide.