- (i) Use the substitution \(t = \tan \frac { X } { 2 }\) to prove the identity
$$\frac { \sin x - \cos x + 1 } { \sin x + \cos x - 1 } \equiv \sec x + \tan x \quad x \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$
(ii) Use the substitution \(t = \tan \frac { \theta } { 2 }\) to determine the exact value of
$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { 5 } { 4 + 2 \cos \theta } d \theta$$
giving your answer in simplest form.