4 The parametric equations of a curve are
$$x = t - \tan t , \quad y = \ln ( \cos t )$$
for \(- \frac { 1 } { 2 } \pi < t < \frac { 1 } { 2 } \pi\).
- Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot t\).
- Hence find the \(x\)-coordinate of the point on the curve at which the gradient is equal to 2 . Give your answer correct to 3 significant figures.
- The polynomial \(\mathrm { f } ( x )\) is of the form \(( x - 2 ) ^ { 2 } \mathrm {~g} ( x )\), where \(\mathrm { g } ( x )\) is another polynomial. Show that \(( x - 2 )\) is a factor of \(\mathrm { f } ^ { \prime } ( x )\).
- The polynomial \(x ^ { 5 } + a x ^ { 4 } + 3 x ^ { 3 } + b x ^ { 2 } + a\), where \(a\) and \(b\) are constants, has a factor \(( x - 2 ) ^ { 2 }\). Using the factor theorem and the result of part (i), or otherwise, find the values of \(a\) and \(b\). [5]