Differentiate $\sin ^2 x$ with respect to $\cos ^2 x$.

If $\cos x = \frac{1}{\sqrt{1 + t^2}}$ and $\sin y = \frac{t}{\sqrt{1 + t^2}}$,then $\frac{dy}{dx} = $

If $x = 2 \sin \theta - \sin 2 \theta$ and $y = 2 \cos \theta - \cos 2 \theta$ where $\theta \in [0, 2 \pi]$,then find the value of $\frac{d^{2} y}{dx^{2}}$ at $\theta = \pi$.

If $x = \sqrt{2} e^t(\sin t - \cos t)$ and $y = \sqrt{2} e^t(\sin t + \cos t)$,then $\left(\frac{d^2 y}{d x^2}\right)_{t = \pi/4} = $

If $x = a \cos \theta$ and $y = b \sin \theta$,then find the value of $\left[\frac{d^2 y}{d x^2}\right]_{\theta = \frac{\pi}{4}}$.

If $x$ and $y$ are connected parametrically by the equations,without eliminating the parameter,find $\frac{d y}{d x}$:
$x = \frac{\sin^3 t}{\sqrt{\cos 2t}}, y = \frac{\cos^3 t}{\sqrt{\cos 2t}}$