7. (i) Given that \(y = \tan x + 2 \cos x\), find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { \pi } { 4 }\).
(ii) Given that \(x = \tan \frac { 1 } { 2 } y\), prove that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { 1 + x ^ { 2 } }\).
(iii) Given that \(y = \mathrm { e } ^ { - x } \sin 2 x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed in the form \(R \mathrm { e } ^ { - x } \cos ( 2 x + \alpha )\). Find, to 3 significant figures, the values of \(R\) and \(\alpha\), where \(0 < \alpha < \frac { \pi } { 2 }\).