Given that \(y = \ln \cos 2 x\), find \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\).
Use Maclaurin's theorem to show that the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \cos 2 x\) are \(- 2 x ^ { 2 } - \frac { 4 } { 3 } x ^ { 4 }\).
Hence find the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \sec ^ { 2 } 2 x\).