Let $f\left(\theta \right)$$=\frac{1}{{\tan }^9\theta }\left\{(1+\tan \theta {)}^{10}+(2+\tan \theta {)}^{10}+\dots +(20+\tan \theta {)}^{10}\right\}-20\tan \theta$ The left hand limit of $f\left(\theta \right)\text{ as }\theta \to \frac{\pi }{2}$ is

Let $x=\tan \theta .$ Then

$f\left({\tan }^{-1}x\right)=\frac{(1+x{)}^{10}+(2+x{)}^{10}+\dots +(20+x{)}^{10}-20x^{10}}{x^9}$

and $x\to \mathrm{\infty }\text{ as }\theta \to \frac{{\pi }^{-}}{2}$

$\begin{array}{l}\therefore \underset{\theta \to \frac{{\pi }^{-}}{2}}{lim} f\left(\theta \right)=\underset{x\to \mathrm{\infty }}{lim} f\left({\tan }^{-1}x\right)\\ =\underset{x\to \mathrm{\infty }}{lim} \frac{(1+x{)}^{10}+(2+x{)}^{10}+\dots +(20+x{)}^{10}-20x^{10}}{x^9}\\ =\underset{x\to \mathrm{\infty }}{lim} \frac{\sum _{r=1}^{20} (r+x{)}^{10}-x^{10}}{x^9}\\ =\underset{x\to \mathrm{\infty }}{lim} \sum _{r=1}^{20} \frac{{ }^{10}{C}_0{r}^{10}{+}^{10}{C}_1{r}^{9}x+\dots {+}^{10}{C}_9{r}^9}{x^9}\\ =\sum _{r=1}^{20} \underset{x\to \mathrm{\infty }}{lim} \left\{\frac{{10}_{C_0}}{r^{10}}+\frac{{ }^{10}{C}_1{r}^9}{x^8}+\dots {+}^{10}{C}_{9}r\right\}\\ =\sum _{r=1}^{10}{ }^{10}{C}_{9}r=10\left(\sum _{r=1}^{20} r\right)=10\times \frac{20\times 21}{2}=2100\end{array}$