(a) Prove the identity
$$\sec ^ { 2 } x + \sec x \tan x \equiv \frac { 1 + \sin x } { \cos ^ { 2 } x }$$
(b) Hence prove that
$$\sec ^ { 2 } x + \sec x \tan x \equiv \frac { 1 } { 1 - \sin x }$$
By differentiating \(\frac { 1 } { \cos x }\), show that if \(y = \sec x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x\).
Using the results of parts (i) and (ii), find the exact value of
$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { 1 - \sin x } \mathrm {~d} x$$