Question

If $a =\displaystyle\lim _{ n \rightarrow \infty} \sum_{ k =1}^{ n } \frac{2 n }{ n ^2+ k ^2}$ 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. $f \left(\frac{ a }{2}\right)=\sqrt{2} f ^{\prime}\left(\frac{ a }{2}\right)$

Answer 1

Answer: (C)

$\sqrt{2} f \left(\frac{ a }{2}\right)= f ^{\prime}\left(\frac{ a }{2}\right)$

Answer 2

The correct option is (C): $\sqrt{2} f \left(\frac{ a }{2}\right)= f ^{\prime}\left(\frac{ a }{2}\right)$