Prove that the equation of the normal at a point of the curve \[x=n\cos t-\cos nt,\qquad y=n\sin t-\sin nt,\] where \(n\) is an integer greater than unity, is \[x\cos \frac{n+1}{2}t + y\sin \frac{n+1}{2}t=(n-1)\cos \frac{n-1}{2}t.\] Show that, if \(n\) is an even integer, the normals at the points \(t\) and \(t+\pi\) are perpendicular and intersect on the circle \[x^2+y^2=(n-1)^2.\]