If \(x = a\sin^{3}t,\ y = b\cos^{3}t\), then find \(\frac{dy}{dx}\) at \(t = \frac{\pi}{4}\).
If \( x = 3\sin t - \sin 3t,\ y = 3\cos t - \cos 3t \), find \( \dfrac{dy}{dx} \) and prove that \( \dfrac{d^2y}{dx^2} = \dfrac{-\operatorname{cosec}^3 2t \cdot \operatorname{cosec} t}{3} \).
If \(x = \cos t\), \(y = \cos mt\), prove that \((1-x^{2})\dfrac{d^{2}y}{dx^{2}} - x\dfrac{dy}{dx} + m^{2}y = 0\).