Question 5 - A-Level Maths

By differentiating \(\frac { 1 } { \cos x }\), show that if \(y = \sec x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x\).
Show that \(\frac { 1 } { \sec x - \tan x } \equiv \sec x + \tan x\).
Deduce that \(\frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \equiv 2 \sec ^ { 2 } x - 1 + 2 \sec x \tan x\).
Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } ( 8 \sqrt { } 2 - \pi )\).