For any positive integer n, let S_n : (0, $$\infty$$) $$\to$$ R be defined by $${S_n}(x) = \sum\nolimits_{k = 1}^n {{{\cot }^{ - 1}}\left( {{{1 + k(k + 1){x^2}} \over x}} \right)} $$, where for any x $$\in$$ R, $${\cot ^{ - 1}}(x) \in (0,\pi )$$ and $${\tan ^{ - 1}}(x) \in \left( { - {\pi \over 2},{\pi \over 2}} \right)$$. Then which of the following statements is (are) TRUE?

For any positive integer n, define

$${f_n}:(0,\infty ) \to R$$ as

$${f_n} = \sum\limits_{j = 1}^n {{{\tan }^{ - 1}}} \left( {{1 \over {1 + (x + j)(x + j - 1)}}} \right)$$

for all x$$ \in $$(0, $$\infty $$). (Here, the inverse trigonometric function tan^$$-$$1 x assumes values in $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$). Then, which of the following statement(s) is (are) TRUE?

If $$\alpha $$ $$ = 3{\sin ^{ - 1}}\left( {{6 \over {11}}} \right)$$ and $$\beta = 3{\cos ^{ - 1}}\left( {{4 \over 9}} \right),$$ where the inverse trigonimetric functions take only the principal values, then the correct options(s) is (are)