The curve $C$ is defined by the parametric equations
$x=\theta+\frac{1}{2} \sin 2 \theta, y=2 \tan \theta \quad \text { where }-\frac{\pi}{2}<\theta<\frac{\pi}{2} \text {. }$
(i) Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1}{\cos ^k \theta}$, where $k$ is an integer to be determined.

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The lines $T$ and $N$ are the tangent and normal to $C$ at the point where $\theta=\frac{\pi}{4}$ respectively.
(ii) Find the equations of $T$ and $N$, leaving your answers in exact form.

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(iii) Find the area enclosed by $T, N$ and the $y$-axis.

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