If $x=\operatorname{cosec} \theta-\sin \theta$, $y=\operatorname{cosec}^{2022} \theta-\sin ^{2022} \theta$ and $\left(\frac{d y}{d x}\right)^2=\frac{k\left(y^2+4\right)}{g(x)}$ where $k \in R$, then $10+k-g(2022)=$

If $x=a\left\{\cos \theta+\log \tan \left(\frac{\theta}{2}\right)\right\}$ and $y=a \sin \theta$, then $\frac{dy}{dx}$ is equal to

Let $t \in (0, 1)$ and $\alpha \in (0, \frac{\pi}{4})$. If $x = \text{cosec}^{-1} \left( \frac{1 + t^2}{2t} \right)$, $y = \cot^{-1} \left( \frac{\sqrt{1 - t^2}}{t} \right)$ and $\frac{dy}{dx} = f(t)$, then the value of $f(\tan \alpha)$ is

If $\tan x = \frac{2t}{1-t^2}$ and $\sin y = \frac{2t}{1+t^2}$,then the value of $\frac{dy}{dx}$ is

Consider the curve represented parametrically by the equations $x = t^3 - 4t^2 - 3t$ and $y = 2t^2 + 3t - 5$,where $t \in \mathbb{R}$. If $H$ denotes the number of points on the curve where the tangent is horizontal and $V$ denotes the number of points where the tangent is vertical,then:

The derivative of $f(\tan x)$ with respect to $g(\sec x)$ at $x = \frac{\pi}{4}$,where $f^{\prime}(1) = 2$ and $g^{\prime}(\sqrt{2}) = 4$,is: