Question:
If $\operatorname{cosec} x+\cot x=\frac{11}{2}$, then the value of $\tan \mathrm{x}$ is ____________ .
Solution:
$\operatorname{cosec} x+\cot x=\frac{11}{2}$ ....(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 $\tan x=\frac{44}{117}$