A curve has parametric equations \(x=5a\sec\theta,\,y=3a\tan\theta\), where \(-\frac{1}{2}\pi < \theta < \frac{1}{2}\pi\) and \(a\) is a positive constant. Find the coordinates of the point on the curve at which the normal is parallel to the line \(y=x\).

How can we find the derivative \(\dfrac{dy}{dx}\) for this curve?

We want the normal to have the same gradient as \(y=x\).

If we know the value of \(\sin\theta\), how can we find the corresponding value of \(\cos\theta\)?