Question

If $\cos ecA+cotA=\frac{11}{2}$, then tan A =
(a) $\frac{21}{22}$

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

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

(d) $\frac{117}{43}$

Answer 1

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

$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ \mathrm{cosec}\mathrm{A}+\cot \mathrm{A}=\frac{11}{2}\left(1\right) \\ \Rightarrow \frac{1}{\mathrm{cosec}\mathrm{A}+\cot \mathrm{A}}=\frac{2}{11} \\ \Rightarrow \frac{{\mathrm{cosec}}^2\mathrm{A}-{\cot }^2\mathrm{A}}{\mathrm{cosec}\mathrm{A}+\cot \mathrm{A}}={\displaystyle \frac{2}{11}} \\ \Rightarrow \frac{\left(\mathrm{cosec}\mathrm{A}+\cot \mathrm{A}\right)\left(\mathrm{cosec}\mathrm{A}-\cot \mathrm{A}\right)}{\left(\mathrm{cosec}\mathrm{A}+\cot \mathrm{A}\right)}={\displaystyle \frac{2}{11}} \\ \therefore \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{A}-\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{A}=\frac{2}{11}\left(2\right) \\ \mathrm{Subtracting}\left(2\right)\mathrm{from}\left(1\right): \\ 2\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{A}=\frac{11}{2}-\frac{2}{11} \\ \Rightarrow 2\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{A}=\frac{121-4}{22} \\ \Rightarrow 2\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{A}=\frac{117}{22} \\ \Rightarrow \mathrm{c}\mathrm{o}\mathrm{t}\mathrm{A}=\frac{117}{44} \\ \Rightarrow \frac{1}{\tan \mathrm{A}}=\frac{117}{44} \\ \Rightarrow \tan \mathrm{A}=\frac{44}{117} \end{gathered}$$