Question:
Considering only the principal values of inverse functions,
the set $A=\left\{x \geq 0: \tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}\right\}$
Correct Option: , 3
Solution:
Consider, $\tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}$
$\Rightarrow \quad \tan ^{-1}\left(\frac{5 x}{1-6 x^{2}}\right)=\frac{\pi}{4}$
$\Rightarrow \frac{5 x}{1-6 x^{2}}=1 \Rightarrow 5 x=1-6 x^{2}$
$\Rightarrow \quad 6 x^{2}+5 x-1=0$
$\Rightarrow \quad(6 x-1)(x+1)=0$
$\Rightarrow \quad x=\frac{1}{6}($ as $x \geq 0)$
Therefore, $A$ is a singleton set.
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