5. (a) Using the identity \(\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B\), prove that $$\cos 2 A \equiv 1 - 2 \sin ^ { 2 } A$$ (b) Show that $$2 \sin 2 \theta - 3 \cos 2 \theta - 3 \sin \theta + 3 \equiv \sin \theta ( 4 \cos \theta + 6 \sin \theta - 3 )$$ (c) Express \(4 \cos \theta + 6 \sin \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
(d) Hence, for \(0 \leq \theta < \pi\), solve $$2 \sin 2 \theta = 3 ( \cos 2 \theta + \sin \theta - 1 )$$ giving your answers in radians to 3 significant figures, where appropriate.
Hence, for \(0 \leq \theta < \pi\), solve \includegraphics[max width=\textwidth, alt={}]{933ec0b9-3496-455e-9c2c-2612e84f63ff-02_20_26_1509_239} giving your answers in radians to 3 significant figures, where appropriate.

5. (a) Using the identity $\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$, prove that

$$\cos 2 A \equiv 1 - 2 \sin ^ { 2 } A$$

(b) Show that

$$2 \sin 2 \theta - 3 \cos 2 \theta - 3 \sin \theta + 3 \equiv \sin \theta ( 4 \cos \theta + 6 \sin \theta - 3 )$$

(c) Express $4 \cos \theta + 6 \sin \theta$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$.\\
(d) Hence, for $0 \leq \theta < \pi$, solve

$$2 \sin 2 \theta = 3 ( \cos 2 \theta + \sin \theta - 1 )$$

giving your answers in radians to 3 significant figures, where appropriate.\\
Hence, for $0 \leq \theta < \pi$, solve\\
\includegraphics[max width=\textwidth, alt={}]{933ec0b9-3496-455e-9c2c-2612e84f63ff-02_20_26_1509_239} giving your answers in radians to 3 significant figures, where appropriate.\\

\hfill \mbox{\textit{Edexcel C3  Q5 [8]}}