Standard +0.3 Part (a) is a straightforward proof requiring direct expansion of standard compound angle formulae and simple algebraic manipulation. Part (b) requires applying the product-to-sum formula, using reciprocal identities, and solving a trigonometric equation within a given interval—standard C3 techniques with multiple steps but no novel insight required. Slightly above average due to the multi-step nature and need to connect parts (a) and (b).
2. (a) Use the identities for \(\cos ( A + B )\) and \(\cos ( A - B )\) to prove that
$$2 \cos A \cos B \equiv \cos ( A + B ) + \cos ( A - B ) .$$
(b) Hence, or otherwise, find in terms of \(\pi\) the solutions of the equation
$$2 \cos \left( x + \frac { \pi } { 2 } \right) = \sec \left( x + \frac { \pi } { 6 } \right) ,$$
for \(x\) in the interval \(0 \leq x \leq \pi\).
2. (a) Use the identities for $\cos ( A + B )$ and $\cos ( A - B )$ to prove that
$$2 \cos A \cos B \equiv \cos ( A + B ) + \cos ( A - B ) .$$
(b) Hence, or otherwise, find in terms of $\pi$ the solutions of the equation
$$2 \cos \left( x + \frac { \pi } { 2 } \right) = \sec \left( x + \frac { \pi } { 6 } \right) ,$$
for $x$ in the interval $0 \leq x \leq \pi$.\\
\hfill \mbox{\textit{Edexcel C3 Q2 [9]}}