| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2006 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then solve equation |
| Difficulty | Standard +0.3 This is a straightforward two-part question requiring standard application of addition formulae for tan(A±B) and double angle formula. Part (i) is algebraic manipulation with tan(45°)=1 simplifying nicely, and part (ii) becomes a simple equation tan(2x)=1 after using the identity. The techniques are routine for P2 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals1.05p Proof involving trig: functions and identities |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use \(\tan(A \pm B)\) formula to express LHS in terms of \(\tan x\) | M1 | |
| Obtain \(\frac{\tan x + 1}{1 - \tan x} \cdot \frac{1 - \tan x}{1 + \tan x}\), or equivalent | A1 | |
| Make relevant use of the \(\tan 2A\) formula | M1 | |
| Obtain given answer correctly | A1 | 4 marks |
| (ii) State or imply \(2x = \tan^{-1}(2/2)\) | M1 | |
| Obtain answer \(x = 22\frac{1}{2}°\) | A1 | |
| Obtain answer \(x = 112\frac{1}{2}°\) and no others in range | A1 | 3 marks |
(i) Use $\tan(A \pm B)$ formula to express LHS in terms of $\tan x$ | M1 |
Obtain $\frac{\tan x + 1}{1 - \tan x} \cdot \frac{1 - \tan x}{1 + \tan x}$, or equivalent | A1 |
Make relevant use of the $\tan 2A$ formula | M1 |
Obtain given answer correctly | A1 | 4 marks |
(ii) State or imply $2x = \tan^{-1}(2/2)$ | M1 |
Obtain answer $x = 22\frac{1}{2}°$ | A1 |
Obtain answer $x = 112\frac{1}{2}°$ and no others in range | A1 | 3 marks |4 (i) Prove the identity
$$\tan \left( x + 45 ^ { \circ } \right) - \tan \left( 45 ^ { \circ } - x \right) \equiv 2 \tan 2 x .$$
(ii) Hence solve the equation
$$\tan \left( x + 45 ^ { \circ } \right) - \tan \left( 45 ^ { \circ } - x \right) = 2 ,$$
for $0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P2 2006 Q4 [7]}}