Inverse Trigonometric Functions Question 1
Question 1 - 2024 (27 Jan Shift 2)
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 :
(1) More than 2
(2) 1
(3) 2
(4) 0
Show Answer
Answer (2)
Solution
$\tan ^{-1} x+\tan ^{-1} 2 x=\frac{\pi}{4} ; x>0$
$\Rightarrow \tan ^{-1} 2 \mathrm{x}=\frac{\pi}{4}-\tan ^{-1} \mathrm{x}$
Taking tan both sides
$\Rightarrow 2 x=\frac{1-x}{1+x}$
$\Rightarrow 2 x^{2}+3 x-1=0$
$x=\frac{-3 \pm \sqrt{9+8}}{8}=\frac{-3 \pm \sqrt{17}}{8}$
Only possible $x=\frac{-3+\sqrt{17}}{8}$
