Question

The solution set of the equation ${\tan }^{-1}x-{\cot }^{-1}x={\cos }^{-1}(2-x)$ is

• A. $(0,1)$
• B. $(-1,1)$
• C. $[1,3)$
• D. $(1,3)$

Answer 1

The correct option is C $[1,3)$

Given, ${\tan }^{-1}x-{\cot }^{-1}x={\cos }^{-1}(2-x)x\in [1,3]$

$2{\tan }^{-1}x-\frac{\pi }{2}={\cos }^{-1}(2-x)$ $[\because {\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}]$

$2{\tan }^{-1}x=\frac{\pi }{2}+{\cos }^{-1}(2-x)$
Take $\sin$ on both sides,
$\Rightarrow 2\frac{x}{(1+x^2)}=2-x........2{\tan }^{-1}x=si{n}^{-1}\frac{2x}{1+x^2}$
$\Rightarrow x^3-2x^2+3x-2=0$
$\Rightarrow x=1$
So, set containing solution is $[1,3)$.