Question 7 - A-Level Maths

Prove that if \(y = \frac { 1 } { \cos \theta }\) then \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \sec \theta \tan \theta\).
Prove the identity \(\frac { 1 + \sin \theta } { 1 - \sin \theta } \equiv 2 \sec ^ { 2 } \theta + 2 \sec \theta \tan \theta - 1\).
Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 + \sin \theta } { 1 - \sin \theta } \mathrm { d } \theta\).
\(8 \quad\) Let \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } - 7 x + 4 } { ( 3 x + 2 ) \left( x ^ { 2 } + 5 \right) }\).
Express \(\mathrm { f } ( x )\) in partial fractions.
Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).