| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then solve equation |
| Difficulty | Moderate -0.3 This is a standard C3 question testing double angle formulae and trigonometric identities. Part (i) is routine bookwork proving cos 2x from cos(A+B). Part (ii) requires algebraic manipulation but follows directly from part (i). Part (iii) is a straightforward equation to solve using the previous results. All steps are well-practiced techniques 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}
\item (i) Use the identity for $\cos ( A + B )$ to prove that
\end{enumerate}
$$\cos 2 x \equiv 2 \cos ^ { 2 } x - 1$$
(ii) Prove that, for $\cos x \neq 0$,
$$2 \cos x - \sec x \equiv \sec x \cos 2 x$$
(iii) Hence, or otherwise, find the values of $x$ in the interval $0 \leq x \leq 180 ^ { \circ }$ for which
$$2 \cos x - \sec x \equiv 2 \cos 2 x$$
\hfill \mbox{\textit{OCR C3 Q4 [9]}}