| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | June |
| Marks | 9 |
| Topic | Trig Proofs |
| Type | Solve equation using proven identity |
| Difficulty | Standard +0.3 Part (a) is a straightforward trigonometric identity proof using standard double angle formulas (cos 2θ = 1 - 2sin²θ and sin 2θ = 2sin θ cos θ). Part (b) applies this identity to solve an equation, requiring substitution of sec²x = 1 + tan²x and algebraic manipulation to reach a quadratic in tan²x. While it involves multiple steps, the techniques are standard A-level Further Maths material with no novel insight required, making it slightly easier than average. |
| 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^21.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
\begin{enumerate}[label=(\alph*)]
\item Prove that
$$1 - \cos 2\theta = \tan \theta \sin 2\theta, \quad \theta \neq \frac{(2n + 1)\pi}{2}, \quad n \in \mathbb{Z}$$
[3]
\item Hence solve, for $-\frac{\pi}{2} < x < \frac{\pi}{2}$, the equation
$$(\sec^2 x - 5)(1 - \cos 2x) = 3\tan^2 x \sin 2x$$
Give any non-exact answer to 3 decimal places where appropriate.
[6]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q6 [9]}}