| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Proofs |
| Type | Prove trigonometric identity |
| Difficulty | Standard +0.3 This is a straightforward trigonometric identity proof using the compound angle formula for tan, followed by algebraic simplification to reach sec 2x. The sketch in part (ii) is routine once the identity is proven. While it requires multiple steps and careful algebra, it follows a standard template for such proofs with no novel insight needed, making it slightly easier than average. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05l Double angle formulae: and compound angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| 4(i) | Use correct tan(A±B)formula and express the LHS in terms of tan x | M1 |
| Usingtan45°=1express LHS as a single fraction | A1 | |
| Use Pythagoras or correct double angle formula | M1 | |
| Obtain given answer | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 4(ii) | Show correct sketch for one branch | B1 |
| Both branches correct and nothing else seen in the interval | B1 | |
| Show asymptote at x = 45° | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 4:
--- 4(i) ---
4(i) | Use correct tan(A±B)formula and express the LHS in terms of tan x | M1
Usingtan45°=1express LHS as a single fraction | A1
Use Pythagoras or correct double angle formula | M1
Obtain given answer | A1
4
--- 4(ii) ---
4(ii) | Show correct sketch for one branch | B1
Both branches correct and nothing else seen in the interval | B1
Show asymptote at x = 45° | B1
3
Question | Answer | Marks\begin{enumerate}[label=(\roman*)]
\item Prove the identity $\tan(45° + x) + \tan(45° - x) = 2 \sec 2x$. [4]
\item Hence sketch the graph of $y = \tan(45° + x) + \tan(45° - x)$ for $0° \leqslant x \leqslant 90°$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2017 Q4 [7]}}