If cosec x+cot x=

$\operatorname{cosec} x+\cot x=\frac{11}{2}$   ....(1)

Since $\operatorname{cosec}^{2} x-\cot ^{2} x=1$

$\Rightarrow(\operatorname{cosec} x-\cot x)(\operatorname{cosec} x+\cot x)=1$

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

Subtrating (2) from (1)

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

$2 \cot x=\frac{121-4}{22}=\frac{117}{22}$

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

i.e $\tan x=\frac{44}{117}$