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 :
Solution
<p>$$\begin{aligned}
& \tan ^{-1} x+\tan ^{-1} 2 x=\frac{\pi}{4} ; x>0 \\
& \Rightarrow \tan ^{-1} 2 x=\frac{\pi}{4}-\tan ^{-1} x
\end{aligned}$$</p>
<p>Taking tan both sides</p>
<p>$$\begin{aligned}
& \Rightarrow 2 \mathrm{x}=\frac{1-\mathrm{x}}{1+\mathrm{x}} \\
& \Rightarrow 2 \mathrm{x}^2+3 \mathrm{x}-1=0 \\
& \mathrm{x}=\frac{-3 \pm \sqrt{9+8}}{8}=\frac{-3 \pm \sqrt{17}}{8}
\end{aligned}$$</p>
<p>Only possible $x=\frac{-3+\sqrt{17}}{8}$</p>
About this question
Subject: Mathematics · Chapter: Inverse Trigonometric Functions · Topic: Domain and Range
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