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