| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2020 |
| Session | October |
| Marks | 12 |
| Topic | Reciprocal Trig & Identities |
| Type | Double angle with reciprocal functions |
| Difficulty | Challenging +1.3 Part (a) requires understanding that sec x = 1/cos x, so cos(sec x) = cos(1/cos x), which is non-standard and requires careful analysis of the composite function's behavior including discontinuities. Part (b) involves using symmetry properties of this unusual function. Part (c) is a challenging algebraic manipulation requiring the double angle formula applied iteratively (tan φ, cot 2φ, tan 4φ) with cot φ = 4, demanding careful tracking of multiple substitutions and simplifications. The combination of an unfamiliar composite trigonometric function and extended multi-step algebraic reasoning with double angle formulas places this significantly above average difficulty. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05l Double angle formulae: and compound angle formulae |
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Sketch the graph of $y = \cos \sec x$ for $0 < x < 4\pi$.
[3]
\item It is given that $\cos \sec \alpha = \cos \sec \beta$, where $\frac{1}{2}\pi < \alpha < \pi$ and $2\pi < \beta < \frac{5}{2}\pi$. By using your sketch, or otherwise, express $\beta$ in terms of $\alpha$.
[2]
\end{enumerate}
\item \begin{enumerate}[label=(\roman*)]
\item Write down the identity giving $\tan 2\theta$ in terms of $\tan \theta$.
[1]
\item Given that $\cot \phi = 4$, find the exact value of $\tan \phi \cot 2\phi \tan 4\phi$, showing all your working.
[6]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2020 Q8 [12]}}