Considering only the Principal values of inverse functions, the set $A=\left\{x \geq 0 \left\lvert\, an ^{-1}(2 x)+ an ^{-1}(3 x)=\frac{\pi}{4} ight. ight\}$

Question

Considering only the Principal values of inverse functions, the set $A=\left\{x \geq 0 \left\lvert\, 	an ^{-1}(2 x)+	an ^{-1}(3 x)=\frac{\pi}{4}ight.ight\}$

Aicharya · Accepted Answer

$\begin{aligned} & 	an ^{-1}(2 x)+	an ^{-1}(3 x)=\frac{\pi}{4} \ 	herefore \quad & 	an ^{-1}\left(\frac{2 x+3 x}{1-(2 x)(3 x)}ight)=\frac{\pi}{4}\end{aligned}$ $\begin{array}{ll} 	herefore & \frac{5 x}{1-6 x^2}=	an \left(\frac{\pi}{4}ight) \ 	herefore & 5 x=1-6 x^2 \ 	herefore & 6 x^2+5 x-1=0 \ 	herefore & 6 x^2+6 x-x-1=0 \ 	herefore & (6 x-1)(x+1)=0 \ 	herefore & x=\frac{1}{6} \ldots[\because x \geq 0] \end{array}$ $	herefore \quad$ set A is a singleton set.

Considering only the Principal values of inverse functions, the set $A=\left\{x \geq 0 \left\lvert\, \tan…

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