Question

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

(1) $\frac{3}{17}$

(2) $\frac{2}{17}$

(3) $\frac{5}{17}$

(4) $\frac{6}{17}$ 

Answer 1

SOLUTION :  Since, $\operatorname{cosec}^{-1}\left(\frac{5}{3}\right)=\tan ^{-1}\left(\frac{3}{4}\right)$

$\cot \left(\tan ^{-1} \frac{3}{4}+\tan ^{-1} \frac{2}{3}\right)=\cot \tan ^{-1}\left[\frac{\frac{3}{4}+\frac{2}{3}}{1-\frac{1}{2}}\right]=\cot \tan ^{-1}\left[\frac{\left(\frac{17}{12}\right)}{\left(\frac{1}{2}\right)}\right]=\cot \left[\tan ^{-1}\left(\frac{17}{6}\right)\right]=\frac{6}{17}$ 

Hence Option 4 is Correct.