JEE Math Practice Question

If $$a = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {{{2n} \over {{n^2} + {k^2}}}} $$ and $$f(x) = \sqrt {{{1 - \cos x} \over {1 + \cos x}}} $$, $$x \in (0,1)$$, then :

1. A.$$2\sqrt 2 f\left( {{a \over 2}} \right) = f'\left( {{a \over 2}} \right)$$
2. B.$$f\left( {{a \over 2}} \right)f'\left( {{a \over 2}} \right) = \sqrt 2 $$
3. C.$$\sqrt 2 f\left( {{a \over 2}} \right) = f'\left( {{a \over 2}} \right)$$
4. D.$$f\left( {{a \over 2}} \right) = \sqrt 2 f'\left( {{a \over 2}} \right)$$