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  4. The value of cot (cosec-1(5/3)+ tan-1(2/3)) is

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

Inverse Trigonometric Functions

Solution:

$cosec^{-1} \frac{5}{3} = sin ^{-1} \frac{3}{5} = tan^{-1} \frac{3}{4}$
$ \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}$