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:

$$\begin{aligned}
y & = \ln \left( \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x \right) \\
& = \ln \left( \sec ^ { 2 } x \right) + \ln \left( \operatorname { cosec } ^ { 2 } x \right) \\
& = 2 \ln \sec x + 2 \ln \operatorname { cosec } x \\
\frac { \mathrm {~d} y } { \mathrm {~d} x } & = 2 \tan x - 2 \cot x \\
& = \frac { 2 \left( \sin ^ { 2 } x - \cos ^ { 2 } x \right) } { \sin x \cos x } \\
& = \frac { - 2 \cos 2 x } { \frac { 1 } { 2 } \sin 2 x } \\
& = - 4 \cot 2 x
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Identify the error in this attempt at a proof.
\item Give a correct version of the proof.
\item 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.
\item 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
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2002 Q7 [18]}}