| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2020 |
| Session | October |
| Marks | 7 |
| Topic | Trig Proofs |
| Type | Solve equation using proven identity |
| Difficulty | Standard +0.3 Part (i) requires applying compound angle formulas and double angle identities to prove a trigonometric identity—standard A-level technique with multiple steps but no novel insight. Part (ii) is a straightforward application of the proven identity with simple algebraic manipulation and solving a quadratic in sin θ. The question is slightly easier than average due to the guided structure and routine methods, though it does require careful algebraic manipulation across 7 marks total. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
\begin{enumerate}[label=(\roman*)]
\item Prove that
$$\cos^2(\theta + 45°) - \frac{1}{2}(\cos 2\theta - \sin 2\theta) \equiv \sin^2 \theta.$$
[4]
\item Hence solve the equation
$$6\cos^2(\frac{1}{2}\theta + 45°) - 3(\cos \theta - \sin \theta) = 2$$
for $-90° < \theta < 90°$.
[3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2020 Q10 [7]}}