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Mathematics Question on Inverse Trigonometric Functions

If tan⁡−1(2x)+tan⁡−1(3x)=π4\tan^{-1} \left(2x\right) + \tan^{-1} \left(3x\right) = \frac{\pi}{4} then x is equal to :

A

-1

B

-2

C

1

D

2

Answer

-1

Explanation

Solution

Given : tan⁡−1(2x)+tan⁡−1(3x)=π4\tan^{-1} \left(2x\right) +\tan^{-1}\left(3x\right) = \frac{\pi}{4} ⇒tan⁡−1(2x+3x)(1−2x3x)=tan⁡−1(1) \Rightarrow \tan^{-1} \frac{\left(2x +3x\right)}{\left(1-2x 3x\right) } = \tan^{-1}\left(1\right) ⇒5x1−6x2=1⇒6x2+5x−1=0 \Rightarrow \frac{5x}{ 1 - 6x^{2}} = 1 \Rightarrow 6x^{2} + 5x - 1 = 0 ⇒(6x−1)(x+1)=0⇒x=16 \Rightarrow \left(6x -1\right) \left(x + 1\right) = 0 \Rightarrow x = \frac{1}{6} or −1- 1 But x = - 1 is in the option .