| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Compound angle with reciprocal functions |
| Difficulty | Standard +0.3 This is a straightforward trigonometric identity question requiring knowledge of sec²θ = 1 + tan²θ and the tan(α+β) formula. Part (i) reduces to a simple linear equation after substitution, and part (ii) is direct formula application. The algebraic manipulation is routine with no conceptual challenges beyond standard C3 content. |
| 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 formulae |
The angles $\alpha$ and $\beta$ are such that
$$\tan \alpha = m + 2 \quad \text{and} \quad \tan \beta = m,$$
where $m$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item Given that $\sec^2 \alpha - \sec^2 \beta = 16$, find the value of $m$. [3]
\item Hence find the exact value of $\tan(\alpha + \beta)$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2009 Q3 [6]}}