7.
$$y = \tan ^ { 2 } x , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$
- Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 \sec ^ { 4 } x - 4 \sec ^ { 2 } x\)
- Hence show that \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = 8 \sec ^ { 2 } x \tan x \left( A \sec ^ { 2 } x + B \right)\), where \(A\) and \(B\) are constants to be found.
- Find the Taylor series expansion of \(\tan ^ { 2 } x\), in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\), up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 3 }\)