question_answer

The value of \[\sum\limits_{k=1}^{13}{\frac{1}{\sin \,\left( \frac{\pi }{4}+\frac{(k-1)\pi }{6} \right)\,\,\sin \,\,\left( \frac{\pi }{4}+\frac{k\pi }{6} \right)}}\]  is equal to

A)  \[3-\sqrt{3}\]                

B)  \[2\,(3-\sqrt{3})\]

C)  \[2(\sqrt{3}-1)\]            

D)  \[2\,(2+\sqrt{3})\]

E)  None of these