\section*{3. For ALL APPLICANTS.}
Note that the arguments of all trigonometric functions in this question are given in terms of degrees. You are not expected to differentiate such a function. The notation $\cos ^ { n } x$ means $( \cos x ) ^ { n }$ throughout.\\
(i) $[ 1$ mark $]$ Without differentiating, write down the maximum value of $\cos \left( 2 x + 30 ^ { \circ } \right)$.\\[0pt]
(ii) [4 marks] Again without differentiating, find the maximum value of

$$\cos \left( 2 x + 30 ^ { \circ } \right) \left( 1 - \cos \left( 2 x + 30 ^ { \circ } \right) \right)$$

(iii) [4 marks] Hence write down the maximum value of

$$\cos ^ { 5 } \left( 2 x + 30 ^ { \circ } \right) \left( 1 - \cos \left( 2 x + 30 ^ { \circ } \right) \right) ^ { 5 }$$

(iv) [6 marks] Find the maximum value of

$$\left( 1 - \cos ^ { 2 } \left( 3 x - 60 ^ { \circ } \right) \right) ^ { 4 } \left( 3 - \cos \left( 150 ^ { \circ } - 3 x \right) \right) ^ { 8 }$$