The derivative of $\sin^2 x$ with respect to $\cos^2 x$ is

If $x=a(\theta+\sin \theta)$ and $y=a(1-\cos \theta)$,then $\left(\frac{d^2 y}{dx^2}\right)_{\theta=\pi / 2}=$

If $x=\sec \theta, y=\tan \theta$,then the value of $\frac{d^{2} y}{d x^{2}}$ at $\theta=\frac{\pi}{4}$ is

For a positive constant $a$,find $\frac{dy}{dx}$,where $y = a^{t+\frac{1}{t}}$ and $x = \left(t+\frac{1}{t}\right)^{a}$.

If $x = at^2$ and $y = 2at$,then find $\frac{dy}{dx}$.

If $x$ and $y$ are connected parametrically by the equations,without eliminating the parameter,find $\frac{dy}{dx}$ for $x = \cos \theta - \cos 2\theta$ and $y = \sin \theta - \sin 2\theta$.