$I=\int \frac{\operatorname{cosec}^2 x-2022}{\cos ^{2022} x} d x$
$\begin{aligned} & =\int(\cos x)^{-2022} \cdot \operatorname{cosec}^2 x d x-2022 \int \frac{d x}{\cos ^{2022} x} \\ & =(\cos x)^{-2022} \int \operatorname{cosec}^2 x d x\end{aligned}$
$\begin{array}{r}\left.\left.-\int[-2022)(\cos x)^{-2023}(-\sin x)\right] \int \operatorname{cosec}^2 x d x\right] d x \\ -2022 \int \frac{d x}{\cos ^{2022} x}\end{array}$
$\begin{aligned} & =(\cos x)^{-2022} \cdot(-\cot x)-2022 \\ & \int(\cos x)^{-2022}(\tan x)(-\cot x) d x-2022 \int \frac{d x}{\cos ^{2022} x}\end{aligned}$
$\begin{aligned} & =\frac{-\cot x}{(\cos x)^{2022}}+2022 \int \frac{d x}{(\cos x)^{2022}}-2022 \\ & \int \frac{d x}{(\cos x)^{2022}}+C=\frac{-\cot x}{(\cos x)^{2022}}+C\end{aligned}$
$\begin{aligned} & \therefore f(x)=\frac{-\cot x}{(\cos x)^{2022}} \\ & \int\left(\frac{\pi}{4}\right)=\frac{-1}{(1 / \sqrt{2})^{2022}}=-2^{1011}\end{aligned}$