| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| Topic | Trig Proofs |
| Type | Solve equation using proven identity |
| Difficulty | Standard +0.3 This is a structured multi-part trigonometric identity question with clear signposting. Part (i) requires standard algebraic manipulation using double angle formulas and difference of squares (both routine A-level techniques), while part (ii) is a straightforward equation solve using the proven result. The 'hence' structure guides students through each step, making it slightly easier than average despite involving multiple trigonometric concepts. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
**Part (i)**
- Write the bracketed expression in terms of sin and cos $\left(\dfrac{\cos^2\theta}{\sin^2\theta} - \dfrac{\sin^2\theta}{\cos^2\theta}\right)$: M1
- Sight of $\sin^2 2\theta = k\sin^2\theta\cos^2\theta$: M1
- Obtain $4(\cos^4\theta - \sin^4\theta)$ AG: A1
- Factorise $\cos^4\theta - \sin^4\theta$: M1
- State explicitly $\cos^2\theta + \sin^2\theta = 1$ to obtain $4\cos 2\theta$ AG: A1 **[5]**
**Part (ii)**
- Divide by 4 and $\cos^{-1}$ in correct order for at least one angle: M1
- Divide angles by 2: M1
- Obtain two angles from correct working: A1
- Obtain 30, 150, 210 and 330: A1 **[4]**
**[Total: 9]**10 (i) Prove that
$$\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 4 \left( \cos ^ { 4 } \theta - \sin ^ { 4 } \theta \right)$$
and hence show that
$$\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 4 \cos 2 \theta$$
(ii) Hence solve the equation $\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 2$ for $0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2012 Q10 [9]}}