Question

The value of $\cot \left(cosec^{-1}\frac{5}{3}+\tan^{-1}\frac{2}{3}\right)$ is

• A. $\frac{5}{17}$
• B. $\frac{6}{17}$
• C. $\frac{3}{17}$
• D. $\frac{4}{17}$

Answer 1

Answer: (B)

$\frac{6}{17}$

Answer 2

$cosec^{-1} \frac{5}{3} = sin ^{-1} \frac{3}{5} = tan^{-1} \frac{3}{4}$ $\therefore cot\left(cosec^{-1} \frac{5}{3} +tan^{-1} \frac{2}{3}\right)$ $= cot \left(tan^{-1} \frac{3}{4} + tan^{-1} \frac{2}{3}\right)$ $= cot\left( tan^{-1} \frac{ \frac{3}{4}+\frac{2}{3}}{1- \frac{3}{4} \times \frac{2}{3}}\right)$ $= cot \left(tan^{-1} \frac{\frac{17}{12}}{\frac{1}{2}}\right)$ $= cot(tan^{-1} \frac {17}{6})$ $= cot\left(cot^{-1} \frac{6}{17}\right)$ $= \frac{6}{17}$