Write down the first three terms of the binomial expansion of \(( 1 + y ) ^ { - 1 }\), in ascending powers of \(y\).
By using the expansion
$$\cos x = 1 - \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 4 } } { 4 ! } - \ldots$$
and your answer to part (a)(i), or otherwise, show that the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\sec x\) are
$$1 + \frac { x ^ { 2 } } { 2 } + \frac { 5 x ^ { 4 } } { 24 }$$
By using Maclaurin's theorem, or otherwise, show that the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\tan x\) are
$$x + \frac { x ^ { 3 } } { 3 }$$
Hence find \(\lim _ { x \rightarrow 0 } \left( \frac { x \tan 2 x } { \sec x - 1 } \right)\).