If \( a=\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{2 n}{n^{2}+k^{2}} \) and \( f(x)=\sqrt...
If \( a=\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{2 n}{n^{2}+k^{2}} \) and \( f(x)=\sqrt{\frac{1-\cos x}{1+\cos x}}, x \in(0,1) \), then
(a) \( 2 \sqrt{2} f\left(\frac{a}{2}\right)=f^{\prime}\left(\frac{a}{2}\right) \)
(b) \( f\left(\frac{a}{2}\right) f^{\prime}\left(\frac{a}{2}\right)=\sqrt{2} \)
(c) \( \sqrt{2} f\left(\frac{a}{2}\right)=f^{\prime}\left(\frac{a}{2}\right) \)
(d) \( f\left(\frac{a}{2}\right)=\sqrt{2} f^{\prime}\left(\frac{a}{2}\right) \)
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