Additional Mathematics 4037/11 — May/June 2011

Walkthrough

The left-hand side is a sum of two fractions with different denominators, so the first move is to put them over a common denominator $(1 - \cos\theta)(1 + \cos\theta)$. The numerator becomes $(1 + \cos\theta) + (1 - \cos\theta)$, where the cosine terms cancel, leaving just $2$. The denominator is a difference of two squares: $(1 + \cos\theta)(1 - \cos\theta) = 1 - \cos^2\theta$. Then the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ rearranged gives $1 - \cos^2\theta = \sin^2\theta$, so the expression becomes $\dfrac{2}{\sin^2\theta}$. Since $\operatorname{cosec}\theta = \dfrac{1}{\sin\theta}$, this is exactly $2\operatorname{cosec}^2\theta$, matching the printed target.

Key Takeaways

• To add fractions, use the product of the denominators as the common denominator.
• Recognise the difference of two squares pattern $(a+b)(a-b) = a^2 - b^2$.
• Rearranging $\sin^2\theta + \cos^2\theta = 1$ lets you swap between $1 - \cos^2\theta$ and $\sin^2\theta$.
• Reciprocal definitions such as $\operatorname{cosec}\theta = \dfrac{1}{\sin\theta}$ convert a fraction into cosec form.

Common Mistakes

• Sign slips when combining the numerators — it must be $(1 + \cos\theta) + (1 - \cos\theta)$, giving $2$, not $2\cos\theta$ or $0$.
• Writing the denominator as $1 + \cos^2\theta$ instead of $1 - \cos^2\theta$ (forgetting the minus from the difference of two squares).
• Using the wrong Pythagorean rearrangement, e.g. writing $1 - \cos^2\theta = -\sin^2\theta$.
• Stopping at $\dfrac{2}{\sin^2\theta}$ without converting to $\operatorname{cosec}^2\theta$ — the final A1 requires the exact printed form.
• Starting from the right-hand side and working backwards; for an AG part you must derive forward to the target.

Things to Be Careful About

• This is a "show that" (AG) question: every algebraic line between the given expression and the target must appear — the mark scheme awards M1 for dealing with the fractions, M1 for the simplification and use of $1 - \cos^2\theta = \sin^2\theta$, and A1 only for reaching $2\operatorname{cosec}^2\theta$.
• The answer must be reached by valid forward working; no marks are given for the correct line obtained from incorrect steps.
• Keep all trigonometric functions in the notation of the paper ($\operatorname{cosec}$, not csc).