Considering only the principal values of inverse trigonometric functions, the number of po…

Question

Considering only the principal values of inverse trigonometric functions, the number of positive real values of $x$ satisfying $$\tan ^{-1}(x)+\tan ^{-1}(2 x)=\frac{\pi}{4}$$ is :

Accepted Answer

$$\begin{aligned}
& \tan ^{-1} x+\tan ^{-1} 2 x=\frac{\pi}{4} ; x>0 \
& \Rightarrow \tan ^{-1} 2 x=\frac{\pi}{4}-\tan ^{-1} x
\end{aligned}$$

Taking tan both sides

$$\begin{aligned}
& \Rightarrow 2 \mathrm{x}=\frac{1-\mathrm{x}}{1+\mathrm{x}} \
& \Rightarrow 2 \mathrm{x}^2+3 \mathrm{x}-1=0 \
& \mathrm{x}=\frac{-3 \pm \sqrt{9+8}}{8}=\frac{-3 \pm \sqrt{17}}{8}
\end{aligned}$$

Only possible $$x=\frac{-3+\sqrt{17}}{8}$$

Solutions · 1