Question

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

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

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

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

Answer 1

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

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