1. Tardigrade
  2. Question
  3. Mathematics
  4. The sum of possible values of x for tan -1( x +1)+ cot -1((1/ x -1))= tan -1((8/31)) is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. The sum of possible values of $x$ for $\tan ^{-1}( x +1)+\cot ^{-1}\left(\frac{1}{ x -1}\right)=\tan ^{-1}\left(\frac{8}{31}\right)$ is

JEE MainJEE Main 2021Inverse Trigonometric Functions

Solution:

$\tan ^{-1}(x+1)+\cot ^{-1}\left(\frac{1}{x-1}\right)=\tan ^{-1} \frac{8}{31}$
Taking tangent both sides :-
$\frac{( x +1)+( x -1)}{1-\left( x ^{2}-1\right)}=\frac{8}{31}$
$\Rightarrow \frac{2 x }{2- x ^{2}}=\frac{8}{31}$
$\Rightarrow 4 x^{2}+31 x-8=0$
$\Rightarrow x =-8, \frac{1}{4}$
But, if $x=\frac{1}{4}$
$\tan ^{-1}(x+1) \in\left(0, \frac{\pi}{2}\right)$
$\& \cot ^{-1}\left(\frac{1}{x-1}\right) \in\left(\frac{\pi}{2}, \pi\right)$
$\Rightarrow LHS >\frac{\pi}{2} \& RHS <\frac{\pi}{2}$
(Not possible)
Hence, $x=-8$