Standard +0.3 This is a standard C3 question with routine techniques: part (a) uses double angle formulae mechanically to derive cos 3x; part (b)(i) is algebraic manipulation of a trig identity using common denominators; part (b)(ii) applies the proven identity to solve a straightforward equation. All steps are textbook exercises requiring no novel insight, making it slightly easier than average.
6. (a) Use the double angle formulae and the identity
$$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$$
to obtain an expression for \(\cos 3 x\) in terms of powers of \(\cos x\) only.
(b) (i) Prove that
$$\frac { \cos x } { 1 + \sin x } + \frac { 1 + \sin x } { \cos x } \equiv 2 \sec x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 }$$
(ii) Hence find, for \(0 < x < 2 \pi\), all the solutions of
$$\frac { \cos x } { 1 + \sin x } + \frac { 1 + \sin x } { \cos x } = 4$$
6. (a) Use the double angle formulae and the identity
$$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$$
to obtain an expression for $\cos 3 x$ in terms of powers of $\cos x$ only.\\
(b) (i) Prove that
$$\frac { \cos x } { 1 + \sin x } + \frac { 1 + \sin x } { \cos x } \equiv 2 \sec x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 }$$
(ii) Hence find, for $0 < x < 2 \pi$, all the solutions of
$$\frac { \cos x } { 1 + \sin x } + \frac { 1 + \sin x } { \cos x } = 4$$
\hfill \mbox{\textit{Edexcel C3 2008 Q6 [11]}}