Considering only the principal values of inverse trigonometric functions, the number of positive real values of satisfying is

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Solution

Q.
Considering only the principal values of inverse trigonometric functions, the number of positive real values of $x$ satisfying $\tan^{- 1} (x) + \tan^{- 1} (2 x) = \frac{π}{4}$ is: [2024]

1 1

2 more than 2

3 2

4 0

Ans.
(1)

Given, $\tan^{- 1} (x) + \tan^{- 1} (2 x) = \frac{π}{4}$

$\Rightarrow \tan^{- 1} (\frac{3 x}{1 - 2 x^{2}}) = \frac{π}{4}$ $(∵ \tan^{- 1} x + \tan^{- 1} y = \tan^{- 1} (\frac{x + y}{1 - x y}))$

$\Rightarrow \frac{3 x}{1 - 2 x^{2}} = \tan \frac{π}{4} \Rightarrow \frac{3 x}{1 - 2 x^{2}} = 1 \Rightarrow 2 x^{2} + 3 x - 1 = 0$

$\Rightarrow x = \frac{- 3 \pm \sqrt{17}}{4}$ $∴$ $Possible value of x = \frac{- 3 + \sqrt{17}}{4}$

Hence, only 1 positive real value of $x$ satisfies the equation.

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