Question

If $\mathrm{cosec}x+\cot x=\frac{11}{2}$, then tan x =
(a) $\frac{21}{22}$

(b) $\frac{15}{16}$

(c) $\frac{44}{117}$

(d) $\frac{117}{44}$

Answer 1

(c) $\frac{44}{117}$

$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ \text{cosec}\mathit{\text{x}}+\text{cot}x=\frac{11}{2}\left(1\right) \\ \Rightarrow \frac{1}{\text{cosec}x+\text{cot}x}=\frac{2}{11} \\ \Rightarrow \frac{{\text{cosec}}^{2}x-{\text{cot}}^{2}x}{\text{cosec}x+\text{cot}x}=\frac{2}{11} \\ \Rightarrow \frac{\left(\text{cosec}x+\text{cot}x\right)\left(\text{cosec}x-\text{cot}x\right)}{\left(\text{cosec}x+\text{cot}x\right)}=\frac{2}{11} \\ \therefore \cos \text{ec}A-\cot x=\frac{2}{11}\left(2\right) \\ \mathrm{Subtracting}\left(2\right)\mathrm{from}\left(1\right): \\ 2\text{cot}x=\frac{11}{2}-\frac{2}{11} \\ \Rightarrow 2\text{cot}x=\frac{121-4}{22} \\ \Rightarrow 2\text{cot}x=\frac{117}{22} \\ \Rightarrow \cot x=\frac{117}{44} \\ \Rightarrow \frac{1}{\tan x}=\frac{117}{44} \\ \Rightarrow \tan x=\frac{44}{117} \end{gathered}$$