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For any positive integer n , define fn...

For any positive integer `n` , define `f_n :(0,oo)rarrR` as `f_n(x)=sum_(j=1)^ntan^(-1)(1/(1+(x+j)(x+j-1)))` for all `x in (0, oo)` . Here, the inverse trigonometric function `tan^(-1)x` assumes values in `(-pi/2,pi/2)dot` Then, which of the following statement(s) is (are) TRUE? `sum_(j=1)^5tan^2(f_j(0))=55` (b) `sum_(j=1)^(10)(1+fj '(0))sec^2(f_j(0))=10` (c) For any fixed positive integer `n` , `(lim)_(xrarroo)tan(f_n(x))=1/n` (d) For any fixed positive integer `n` , `(lim)_(xrarroo)sec^2(f_n(x))=1`

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