9 It is given that \(\mathrm { f } ( \theta ) = \sin \left( \theta + 30 ^ { \circ } \right) + \cos \left( \theta + 60 ^ { \circ } \right)\).
- Show that \(\mathrm { f } ( \theta ) = \cos \theta\). Hence show that
$$f ( 4 \theta ) + 4 f ( 2 \theta ) \equiv 8 \cos ^ { 4 } \theta - 3 .$$
- Hence
(a) determine the greatest and least values of \(\frac { 1 } { \mathrm { f } ( 4 \theta ) + 4 \mathrm { f } ( 2 \theta ) + 7 }\) as \(\theta\) varies,
(b) solve the equation
$$\sin \left( 12 \alpha + 30 ^ { \circ } \right) + \cos \left( 12 \alpha + 60 ^ { \circ } \right) + 4 \sin \left( 6 \alpha + 30 ^ { \circ } \right) + 4 \cos \left( 6 \alpha + 60 ^ { \circ } \right) = 1$$
for \(0 ^ { \circ } < \alpha < 60 ^ { \circ }\).
\section*{END OF QUESTION PAPER}