Answer 1

\[\frac{44}{117}\]

We have:

\[ cosec x + \cot x = \frac{11}{2} \left( 1 \right)\]

\[ \Rightarrow \frac{1}{cosecx + \cot x} = \frac{2}{11}\]

\[ \Rightarrow \frac{{cosec}^2 x - \cot^2 x}{cosecx + \cot x} = \frac{2}{11}\]

\[ \Rightarrow \frac{\left( cosec x + \cot x \right)\left( cosec x - \cot x \right)}{\left( cosec x + \cot x \right)} = \frac{2}{11}\]

\[ \therefore cosec A-\cot x = \frac{2}{11} \left( 2 \right)\]

Subtracting ( 2 ) from ( 1 ): 

\[2\cot x = \frac{11}{2} - \frac{2}{11}\]

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

\[ \Rightarrow 2\cot x = \frac{117}{22}\]

\[ \Rightarrow \cot x = \frac{117}{44}\]

\[ \Rightarrow \frac{1}{\tan x} = \frac{117}{44}\]

\[ \Rightarrow \tan x = \frac{44}{117}\]