question_answer

If \[\mathbf{2 sin }\alpha \mathbf{=}\sqrt{\mathbf{3}}\] then \[\frac{\mathbf{ta}{{\mathbf{n}}^{\mathbf{2}}}\alpha \mathbf{+3}\mathbf{.cot}\alpha }{\mathbf{co}{{\mathbf{t}}^{\mathbf{2}}}\alpha \mathbf{+2tan}\alpha }\] is equal to ______ (where \[\alpha \] is an acute angle)

A)  \[\frac{3(\sqrt{3}+1)}{1+6\sqrt{3}}\]                                                    

B)  \[\frac{-\,45-51\sqrt{3}}{-107}\]

C)  \[\frac{50\sqrt{3}+45}{-106}\]                                                  

D)  \[\frac{-\,51\sqrt{3}}{107}\]

E)  None of these