Question

If $\mathrm{cosec}x+\cot x=\frac{11}{2},$ then the value of tan x is ______________.

Answer 1

$$\begin{gathered} \mathrm{cosec}x+\cot x=\frac{11}{2}\left(1\right) \\ \mathrm{Since}{\mathrm{cosec}}^{2}x-{\cot }^{2}x=1 \\ \Rightarrow \left(\mathrm{cosec}x-\cot x\right)\left(\mathrm{cosec}x+\cot x\right)=1 \\ \Rightarrow \mathrm{cosec}x-\cot x=\frac{2}{11}\left(2\right) \\ \mathrm{Subtrating}\left(2\right)\mathrm{from}\left(1\right) \\ \mathrm{cosec}x+\tan x-\mathrm{cosec}x+\cot x=\frac{11}{2}-\frac{2}{11} \\ 2\cot x=\frac{121-4}{22}=\frac{117}{22} \\ \cot x=\frac{117}{44} \\ \mathrm{i}.\mathrm{e}\tan x=\frac{44}{117} \end{gathered}$$