Additional Mathematics 4037/21 — May/June 2013

Walkthrough

The left-hand side has two fractions with the same denominator $\cos \theta$, squared. Expanding each numerator with $(a \pm b)^2 = a^2 \pm 2ab + b^2$, the middle terms $+2\sin \theta$ and $-2\sin \theta$ cancel when added, leaving $2 + 2\sin^2 \theta$ over $\cos^2 \theta$. The Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ lets us rewrite $2 + 2\sin^2 \theta$ as $4\sin^2 \theta + 2\cos^2 \theta$, which splits into $\frac{4\sin^2 \theta}{\cos^2 \theta} + \frac{2\cos^2 \theta}{\cos^2 \theta}$. The first term is $4\tan^2 \theta$ (since $\tan \theta = \frac{\sin \theta}{\cos \theta}$) and the second is simply $2$, giving exactly the printed right-hand side.

Key Takeaways

• Expanding $(1 + \sin \theta)^2$ and $(1 - \sin \theta)^2$ together makes the cross terms cancel — a very common pattern in identity proofs.
• The Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ is the main tool for converting between $\sin^2$ and $\cos^2$.
• Dividing term by term by $\cos^2 \theta$ converts everything into $\tan^2 \theta$ and constants.

Common Mistakes

• Expanding $(1 \pm \sin \theta)^2$ as $1 \pm \sin^2 \theta$, forgetting the middle term $\pm 2\sin \theta$ — this destroys the proof.
• Cancelling $\sin \theta$ or dividing through by $\cos \theta$ at some stage, which can lose solutions or break the identity's generality.
• Working backwards from the RHS: the mark scheme allows it only if done rigorously; starting forward from the LHS is safest.
• Not showing every intermediate line — the scheme awards B1 for each distinct correct step ($\frac{2 + 2\sin^2 \theta}{\cos^2 \theta}$, the Pythagoras use, the $\tan$ conversion, and completion), so skipped algebra loses marks.

Things to Be Careful About

• This is an "AG" (answer given) style part: you must derive forward to exactly the printed target and show sufficient correct detail for all four marks.
• Keep the angle $\theta$ consistent throughout — the scheme penalises inconsistent notation (recoverable, but avoid it).
• Every mark corresponds to a visible line of working; do not jump straight from the expanded form to $2 + 4\tan^2 \theta$ without showing the Pythagorean substitution.