■ GDC allowed ■(a) (i) The period of the tangent function is \({\rm{\pi }}\). Hence, any two angles \(\theta\) and \(\theta + {\rm{\pi }}\) will have the same tangent value; i.e. \(\tan \left( \theta \right) = \tan \left( {\theta + {\rm{\pi }}} \right)\) for any value of \(\theta\) such that \(\theta \ne \left( {2n + 1} \right)\dfrac{{\rm{\pi }}}{2},\;n \in \mathbb{Z}\).Thus, there are infinite counterexamples to the statement.One counterexample is \(A = \dfrac{{\rm{\pi }}}{3}\) and \(B = \dfrac{{\rm{\pi }}}{3} + {\rm{\pi }} = \dfrac{{4{\rm{\pi }}}}{3}\;\;\; \Rightarrow \) \(\tan \dfrac{{\rm{\pi }}}{3} = \sqrt 3 \) and \(\tan \dfrac{{4{\rm{\pi }}}}{3} = \sqrt 3 \)Therefore, \(\tan \left( A \right) = \tan \left( B \right)\) but \(A \ne B\).(ii) There are three possible situations if angles...