Definite Integration

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 :