$\underset{\mathrm{n}\to \mathrm{\infty }}{\mathrm{Lt}}\left[\frac{1}{{\mathrm{n}}^2}{\sec }^2\frac{1}{{\mathrm{n}}^2}+\frac{2}{{\mathrm{n}}^2}{\sec }^2\frac{4}{{\mathrm{n}}^2}+\dots ..+\frac{1}{\mathrm{n}}{\sec }^{2}1\right]=$

$=\underset{\mathrm{n}\to \mathrm{\infty }}{\mathrm{Lt}}\left[\frac{1}{{\mathrm{n}}^2}{\sec }^2\frac{1}{{\mathrm{n}}^2}+\frac{2}{{\mathrm{n}}^2}{\sec }^2\frac{4}{{\mathrm{n}}^2}+\dots ..+\frac{1}{\mathrm{n}}{\sec }^{2}1\right]$

$$\begin{gathered} =\underset{n\to \mathrm{\infty }}{lim} \frac{r}{n^2}{\sec }^2\frac{r^2}{n^2} \\ =\underset{n\to \mathrm{\infty }}{lim} \frac{1}{n}\cdot \frac{r}{n}{\sec }^2\frac{r^2}{n^2} \\ ={\int }_0^1 x{\sec }^2{x}^{2}dx \\ =\frac{1}{2}\times {\int }_0^1 2x{\sec }^2{x}^{2}dx \\ \text{ Put }{x}^2=t \\ =\frac{1}{2}{\int }_0^1 {\sec }^2\mathrm{tdt}=\frac{1}{2}(\tan t{)}_0^1 \\ =\frac{{\displaystyle 1}}{{\displaystyle 2}}\tan 1 \end{gathered}$$