3.(i)Determine the value of $k$ such that

$$\arctan \frac { 1 } { 2 } - \arctan \frac { 1 } { 3 } = \arctan k$$

(ii)(a)Prove that

$$\cos 3 A \equiv 4 \cos ^ { 3 } A - 3 \cos A$$

Given that $a = \cos 20 ^ { \circ }$\\
(b)write down,in terms of $a$ ,an expression for $\cos 40 ^ { \circ }$\\
(c)determine,in terms of $a$ ,a simplified expression for $\cos 80 ^ { \circ }$\\
(d)Use part(a)to show that

$$4 a ^ { 3 } - 3 a = \frac { 1 } { 2 }$$

(e)Hence,or otherwise,show that

$$\cos 20 ^ { \circ } \cos 40 ^ { \circ } \cos 80 ^ { \circ } = \frac { 1 } { 8 }$$

\hfill \mbox{\textit{Edexcel AEA 2024 Q3 [14]}}