Question

\[ \text{If } \cosec x+\cot x=\frac{11}{2}, \]

\[ \text{then the value of } \tan x \text{ is} \]

Solution

Using identity

\[ (\cosec x+\cot x)(\cosec x-\cot x)=1 \]

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

Subtracting,

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

\[ = \frac{121-4}{22} = \frac{117}{22} \]

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

Therefore,

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

Answer

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