| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 8 |
| Topic | Trig Proofs |
| Type | Solve equation using proven identity |
| Difficulty | Standard +0.8 Part (a) requires algebraic manipulation with reciprocal trig identities and finding a common denominator—moderately technical but standard Further Maths fare. Part (b) requires recognizing how to apply the proven identity, rearranging to use it, then solving a compound angle equation over an extended interval requiring careful consideration of periodicity. The multi-step nature, compound angle work, and need for strategic insight to connect parts (a) and (b) place this above average difficulty but not at the extreme end for Further Maths. |
| 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^21.05o Trigonometric equations: solve in given intervals |
\begin{enumerate}[label=(\alph*)]
\item Prove the identity $\frac{\cos x}{\sec x + 1} + \frac{\cos x}{\sec x - 1} = 2\cot^2 x$ [3 marks]
\item Hence, solve the equation
$$\frac{\cos\left(2\theta + \frac{\pi}{3}\right)}{\sec\left(2\theta + \frac{\pi}{3}\right) + 1} = \cot\left(2\theta + \frac{\pi}{3}\right) - \frac{\cos\left(2\theta + \frac{\pi}{3}\right)}{\sec\left(2\theta + \frac{\pi}{3}\right) - 1}$$
in the interval $0 \leq \theta \leq 2\pi$, giving your values of $\theta$ to three significant figures where appropriate. [5 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q7 [8]}}