| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Topic | Trig Proofs |
| Type | Prove trigonometric identity |
| Difficulty | Challenging +1.2 Part (a) requires proving a non-standard identity involving double angle formulas and reciprocal functions, demanding algebraic manipulation beyond routine exercises. Part (b) involves solving a trigonometric equation with compound angles requiring the application of addition formulas and careful algebraic work. Both parts require multi-step reasoning and are more challenging than typical A-level questions, but don't reach the level of requiring novel geometric insight or extended proof techniques. |
| Spec | 1.05o Trigonometric equations: solve in given intervals1.05p Proof involving trig: functions and identities |
\begin{enumerate}[label=(\alph*)]
\item Prove the following trigonometric identities. You must show all of your algebraic steps clearly.
$$(\cos x + \sin x)(\cos x - \sec x) \equiv 2 \cot 2x$$ [3]
\item Solve the following equation for $x$ in the interval $0 \leq x \leq \pi$.
Giving your answers in terms of $\pi$.
$$\sin\left(2x + \frac{\pi}{6}\right) = \frac{1}{2}\sin\left(2x - \frac{\pi}{6}\right)$$ [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2021 Q6 [7]}}