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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Inverse Trigonometric Functions

NDA 2022 · Mathematics · Inverse Trigonometric Functions

If ^ -1 ((1)/(2))+ ^ -1 ((x)/(3))=(π)/(4) , where 0 < x < 6 , then...

If tan⁡−1(12)+tan⁡−1(x3)=π4\tan ^{-1}\left(\frac{1}{2}\right)+\tan ^{-1}\left(\frac{x}{3}\right)=\frac{\pi}{4}, where 0<x<60 < x < 6, then what is xx equal to?

Options

  1. A1
  2. B2
  3. C3
  4. D5

Correct answer

A. 1

Step-by-step solution

∵tan⁡−1(12)+tan⁡−1(x3)=π4⇒tan⁡−1(x3)=tan⁡−11−tan⁡−1(12)⇒tan⁡−1x3=tan⁡−1(1−121+12)⇒tan⁡−1x3=tan⁡−1(13)⇒x3=13⇒x=1\begin{aligned} & \because \quad \tan ^{-1}\left(\frac{1}{2}\right)+\tan ^{-1}\left(\frac{x}{3}\right)=\frac{\pi}{4} \\ \Rightarrow & \tan ^{-1}\left(\frac{x}{3}\right)=\tan ^{-1} 1-\tan ^{-1}\left(\frac{1}{2}\right) \\ \Rightarrow & \tan ^{-1} \frac{x}{3}=\tan ^{-1}\left(\frac{1-\frac{1}{2}}{1+\frac{1}{2}}\right) \\ \Rightarrow & \tan ^{-1} \frac{x}{3}=\tan ^{-1}\left(\frac{1}{3}\right) \\ \Rightarrow & \frac{x}{3}=\frac{1}{3} \Rightarrow x=1\end{aligned}
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