If $ an ^{-1} 2 x+ an ^{-1} 3 x=\frac{\pi}{4}$, then value of $x$ is

Question

If $	an ^{-1} 2 x+	an ^{-1} 3 x=\frac{\pi}{4}$, then value of $x$ is

Aicharya · Accepted Answer

$\begin{aligned} & 	an ^{-1} 2 x+	an ^{-1} 3 x=\frac{\pi}{4} \ & \Rightarrow 	an ^{-1}\left(\frac{2 x+3 x}{1-2 x 	imes 3 x}ight)=	an ^{-1}(1) \ & \Rightarrow \frac{5 x}{1-6 x^2}=1 \ & \Rightarrow 6 x^2+5 x-1=0 \ & \Rightarrow(6 x-1)(x+1)=0 \ & \Rightarrow x=\frac{1}{6} 	ext { or } x=-1 \ & \Rightarrow x=\frac{1}{6} 	ext { satisfies the equation }\end{aligned}$

If $\tan ^{-1} 2 x+\tan ^{-1} 3 x=\frac{\pi}{4}$, then value of $x$ is

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