| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2013 |
| Session | June |
| Marks | 12 |
| Topic | Trig Proofs |
| Type | Solve equation using proven identity |
| Difficulty | Challenging +1.2 This question requires proving a trigonometric identity using standard double-angle formulas, then applying it to find an exact value and evaluate a definite integral. While it involves multiple steps and some algebraic manipulation, the techniques are standard for A-level/Pre-U: the identity proof follows directly from cosec 2x = 1/sin 2x and cot 2x = cos 2x/sin 2x, the exact value uses the proven identity with substitution, and the integral becomes straightforward once the identity tan²x is recognized. The question is moderately challenging due to the multi-part nature and need for careful algebraic work, but doesn't require exceptional insight beyond applying known formulas systematically. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05l Double angle formulae: and compound angle formulae |
(i) $\operatorname{cosec} 2x = \frac{1}{\sin 2x}$, $\cot 2x = \frac{\cos 2x}{\sin 2x}$ OR $\frac{1}{\tan 2x}$ seen — B1;
$\operatorname{cosec} 2x - \cot 2x = \frac{1 - \cos 2x}{\sin 2x}$ — M1;
$= \frac{1-(1-2\sin^2 x)}{2\sin x \cos x}$ — M1;
$= \frac{2\sin^2 x}{2\sin x \cos x}$ — A1;
$= \frac{\sin x}{\cos x} = \tan x$ — A1;
$\tan\frac{3}{8}\pi = \operatorname{cosec}\frac{3}{4}\pi - \cot\frac{3}{4}\pi = 1 + \sqrt{2}$ — B1 **[6]**
(ii) $\int_{\frac{1}{4}\pi}^{\frac{3}{8}\pi}(\operatorname{cosec} 2x - \cot 2x)^2\,dx = \int_{\frac{1}{4}\pi}^{\frac{3}{8}\pi}\tan^2 x\,dx$ — M1, A1;
$= \int_{\frac{1}{4}\pi}^{\frac{3}{8}\pi}\sec^2 x \pm 1\,dx$ — M1, A1;
$= \left[\tan x - x\right]_{\frac{1}{4}\pi}^{\frac{3}{8}\pi}$ — M1;
$= \sqrt{2} - \frac{1}{8}\pi$ — A1 **[6]**
**Total: [12]**9 (i) Prove that $\operatorname { cosec } 2 x - \cot 2 x \equiv \tan x$ and hence find an exact value for $\tan \left( \frac { 3 } { 8 } \pi \right)$.\\
(ii) Find the exact value of $\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 3 } { 8 } \pi } ( \operatorname { cosec } 2 x - \cot 2 x ) ^ { 2 } \mathrm {~d} x$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2013 Q9 [12]}}