T-20 LIMIT ;)

$the\quad above\quad expression\quad can\quad be\quad written\quad as\\ =\lim _{ \theta \rightarrow \frac { \pi }{ 2 } }{ \frac { { (1+tan\theta ) }^{ 10 }+{ (2+tan\theta ) }^{ 10 }...{ (20+tan\theta ) }^{ 10 }-20{ tan }^{ 10 }\theta }{ { tan }^{ 9 }\theta } } \\ =\lim _{ \theta \rightarrow \frac { \pi }{ 2 } }{ tan\theta \quad [\quad ({ cot\theta +1) }^{ 10 }\quad +\quad (2{ cot\theta +1) }^{ 10 }\quad ...+({ 20cot\theta +1) }^{ 10 }\quad -20] } \\ now\quad on\quad applying\quad binomial\quad approxiamation\quad to\quad each\quad term,we\\ obtain:-\\ =\lim _{ \theta \rightarrow \frac { \pi }{ 2 } }{ tan\theta \quad [\quad ({ 10cot\theta +1) }\quad +\quad (2{ 0cot\theta +1) }\quad ...+({ 200cot\theta +1) }\quad -20] } \\ =\lim _{ \theta \rightarrow \frac { \pi }{ 2 } }{ tan\theta \quad [\quad { 10cot\theta }\quad +\quad { 20cot\theta }\quad ...+{ 200cot\theta }\quad +\quad 20\quad -\quad 20] } \\ =10\sum _{ i=1 }^{ 20 }{ i } \\ =10.(20.21)/2\\ =2100$

I chose 2100 because of the 9+10 meme/vine. I absolutely do not know how to do this.

by binomial approximation method and summation factors: