7．The variable \(y\) is defined by $$y = \ln \left( \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x \right) \text { for } 0 < x < \frac { \pi } { 2 } .$$ A student was asked to prove that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - 4 \cot 2 x .$$ The attempted proof was as follows： $$\begin{aligned} y & = \ln \left( \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x \right)
& = \ln \left( \sec ^ { 2 } x \right) + \ln \left( \operatorname { cosec } ^ { 2 } x \right)
& = 2 \ln \sec x + 2 \ln \operatorname { cosec } x
\frac { \mathrm {~d} y } { \mathrm {~d} x } & = 2 \tan x - 2 \cot x
& = \frac { 2 \left( \sin ^ { 2 } x - \cos ^ { 2 } x \right) } { \sin x \cos x }
& = \frac { - 2 \cos 2 x } { \frac { 1 } { 2 } \sin 2 x }
& = - 4 \cot 2 x \end{aligned}$$ （a）Identify the error in this attempt at a proof．
（b）Give a correct version of the proof．
（c）Find and simplify a general relationship between \(p\) and \(q\) ，where \(p\) and \(q\) are variables that depend on \(x\) ，such that the student would obtain the correct result when differentiating \(\ln ( p + q )\) with respect to \(x\) by the above incorrect method．
（d）Given that \(p ( x ) = k \sec r x\) and \(q ( x ) = \operatorname { cosec } ^ { 2 } x\) ，where \(k\) and \(r\) are positive integers，find the values of \(k\) and \(r\) such that \(p\) and \(q\) satisfy the relationship found in part（c）． \section*{END} Marks for presentation： 7
TOTAL MARKS： 100