Let \(P(t)\) denote the point \[ (\cos t, f(t)\sin t), \] where \(f(t)\) is a strictly positive continuous function of \(t\) in \(0 \le t \le 2\pi\) with \(f(2\pi)=f(0)\); and let \(\mathcal{C}\) be the closed curve described by \(P(t)\) as \(t\) varies from \(0\) to \(2\pi\). Show that the area \(A\) enclosed by \(\mathcal{C}\) is \[ A = \int_0^{2\pi} f(t)\sin^2 t \,dt. \] Find an expression for the area \(T(t_1, t_2, \dots, t_n)\) of the polygon with vertices \(P(t_1), P(t_2), \dots, P(t_n)\), where \[ t_1 < t_2 < \dots < t_n < t_1+2\pi, \] and show that \[ \int_0^{2\pi} T\left(t, t+\frac{2\pi}{n}, t+\frac{4\pi}{n}, \dots, t+(n-1)\frac{2\pi}{n}\right) dt = nA\sin\frac{2\pi}{n}. \] Deduce that \[ T(t_1, t_2, \dots, t_n) \ge \frac{n\sin\frac{2\pi}{n}}{2\pi} A \] for some \(t_1, t_2, \dots, t_n\).