Question

Considering only the principal values of inverse functions, the set $A = \{ x \ge 0 : \tan^{-1} (2x) + \tan^{-1} (3x) = \frac{\pi}{4} \}$

• A. is an empty set
• B. Contains more than two elements
• C. Contains two elements
• D. is a singleton

Answer 1

Answer: (D)

$\tan^{-1}\left(2x\right)+\tan^{-1}\left(3x\right)=\pi/4 $
$\Rightarrow \frac{5x}{1-6x^{2}} = 1$
$ \Rightarrow 6x^{2}+5x-1=0$
$ x=-1 $ or $x=\frac{1}{6} $
$x = \frac{1}{6} \because x >0 $