C3 January 2008 Q6
6.
(a) Use the double angle formulae and the identity\[\cos(A + B) \equiv \cos A\cos B - \sin A\sin B\]to obtain an expression for \(\cos 3x\) in terms of powers of \(\cos x\) only. (4)
(b)
(i) Prove that\[\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x} \equiv 2\sec x, \qquad x \neq (2n + 1)\frac{\pi}{2}.\] (4)
(ii) Hence find, for \(0 < x < 2\pi\), all the solutions of\[\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x} = 4.\] (3)
| Scheme | Marks |
|---|---|
| \(\cos(2x + x) = \cos 2x\cos x - \sin 2x\sin x\) | M1 |
| \(= (2\cos^2 x - 1)\cos x - (2\sin x\cos x)\sin x\) | M1 |
| \(= (2\cos^2 x - 1)\cos x - 2(1 - \cos^2 x)\cos x\) any correct expression | A1 |
| \(= 4\cos^3 x - 3\cos x\) | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| (i) \(\dfrac{\cos x}{1 + \sin x} + \dfrac{1 + \sin x}{\cos x} = \dfrac{\cos^2 x + (1 + \sin x)^2}{(1 + \sin x)\cos x}\) | M1 |
| \(= \dfrac{\cos^2 x + 1 + 2\sin x + \sin^2 x}{(1 + \sin x)\cos x}\) | A1 |
| \(= \dfrac{2(1 + \sin x)}{(1 + \sin x)\cos x}\) | M1 |
| \(= \dfrac{2}{\cos x} = 2\sec x\ \ \ast\) cso | A1 |
| (4) | |
| (ii) \(\sec x = 2\) or \(\cos x = \dfrac{1}{2}\) | M1 |
| \(x = \dfrac{\pi}{3}, \dfrac{5\pi}{3}\) accept awrt 1.05, 5.24 | A1, A1 |
| (3) | |
| (11 marks) |