June 2023 Paper 2 Q8
8
(a) Given that \(\cos\theta \neq \pm 1\), prove the identity\[\frac{1}{1 - \cos\theta} + \frac{1}{1 + \cos\theta} \equiv 2\operatorname{cosec}^2\theta\] [4 marks]
(b) Hence, find the set of values of \(A\) for which the equation\[\frac{1}{1 - \cos\theta} + \frac{1}{1 + \cos\theta} = A\]has real solutions.
Fully justify your answer. [3 marks]
(c) Given that \(\theta\) is obtuse and\[\frac{1}{1 - \cos\theta} + \frac{1}{1 + \cos\theta} = 16\]find the exact value of \(\cot\theta\) [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Recalls \(\operatorname{cosec}\theta = \dfrac{1}{\sin\theta}\) PI by use of \(\operatorname{cosec}^2\theta = \dfrac{1}{\sin^2\theta}\) | B1 | 1.2 |
| Recalls \(\cos^2\theta + \sin^2\theta = 1\) OE | B1 | 1.2 |
| Forms a single fraction with a denominator of \((1 - \cos\theta)(1 + \cos\theta)\) OE | M1 | 1.1a |
| Completes reasoned argument using \(\cos^2\theta + \sin^2\theta = 1\) to prove the given identity. AG | R1 | 2.1 |
| (4) |
Typical solution
\[\frac{1}{1 - \cos\theta} + \frac{1}{1 + \cos\theta}\]\[\equiv \frac{1 + \cos\theta + 1 - \cos\theta}{(1 - \cos\theta)(1 + \cos\theta)}\]\[\equiv \frac{2}{1 - \cos^2\theta}\]\[\equiv \frac{2}{\sin^2\theta}\]\[\equiv 2\operatorname{cosec}^2\theta\]| Scheme | Marks | AO |
|---|---|---|
| Forms the equation \(2\operatorname{cosec}^2\theta = A\) or \(\dfrac{2}{\sin^2\theta} = A\) OE PI by \(A \geqslant 2\) | M1 | 1.1a |
| Explains that \(\operatorname{cosec}\theta \leqslant -1,\ \operatorname{cosec}\theta \geqslant 1\) or \(\operatorname{cosec}^2\theta \geqslant 1\) or Explains that \(-1 \leqslant \sin\theta \leqslant 1\) or \(\sin^2\theta \leqslant 1\) Accept an accurate sketch of \(y = \operatorname{cosec}^2\theta\) with 1 labelled on the \(y\)-axis Condone strict inequalities. | E1 | 2.4 |
| Deduces \(A \geqslant 2\) | R1 | 2.2a |
| (3) |
Typical solution
\[2\operatorname{cosec}^2\theta = A\]\[\operatorname{cosec}\theta \leqslant -1 \text{ or } \operatorname{cosec}\theta \geqslant 1\]\[\text{Hence } \operatorname{cosec}^2\theta \geqslant 1\]\[\therefore A \geqslant 2\]| Scheme | Marks | AO |
|---|---|---|
| Uses the identity from part (a) to obtain \(2\left(1 + \cot^2\theta\right) = 16\) or \(\operatorname{cosec}^2\theta = 8\) or \(\sin^2\theta = \dfrac{1}{8}\) or \(\cos^2\theta = \dfrac{7}{8}\) | M1 | 1.1a |
| Obtains \(\cot^2\theta = 7\) PI by \(\cot\theta = \sqrt{7}\) or \(\cot\theta = -\sqrt{7}\) | A1 | 1.1b |
| Deduces \(\cot\theta = -\sqrt{7}\) | R1 | 2.2a |
| (3) | ||
| (10 marks) |