If cosec x

(c) $\frac{44}{117}$

We have:

$\operatorname{cosec} x+\cot x=\frac{11}{2}$       ...(i)

$\Rightarrow \frac{1}{\operatorname{cosec} x+\cot x}=\frac{2}{11}$

$\Rightarrow \frac{\operatorname{cosec}^{2} x-\cot ^{2} x}{\operatorname{cosec} x+\cot x}=\frac{2}{11}$

$\Rightarrow \frac{(\operatorname{cosec} x+\cot x)(\operatorname{cosec} x-\cot x)}{(\operatorname{cosec} x+\cot x)}=\frac{2}{11}$

$\therefore \operatorname{cosec} A-\cot x=\frac{2}{11}$           ...(ii)

Subtracting (2) from (1):

$2 \cot x=\frac{11}{2}-\frac{2}{11}$

$\Rightarrow 2 \cot x=\frac{121-4}{22}$

$\Rightarrow 2 \cot x=\frac{117}{22}$

$\Rightarrow \cot x=\frac{117}{44}$

$\Rightarrow \frac{1}{\tan x}=\frac{117}{44}$

$\Rightarrow \tan x=\frac{44}{117}$