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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Inverse Trigonometric Functions

NDA 2019 · Mathematics · Inverse Trigonometric Functions

What is \ 2 ^ -1 ((1)/(3))\ equal to?

What is tan⁡\tan\left\{2 \tan ^{-1}\left(\frac{1}{3}\right)\right\} equal to?

Options

  1. A23\frac{2}{3}
  2. B34\frac{3}{4}
  3. C38\frac{3}{8}
  4. D19\frac{1}{9}

Correct answer

B. 34\frac{3}{4}

Step-by-step solution

tan⁡{2tan⁡−1(13)}∵2tan⁡−1x=tan⁡−12x1−x2∴2tan⁡−1(13)=tan⁡−12/31−1/9=tan⁡−1(34)∴tan⁡{2tan⁡−1(13)}=tan⁡(tan⁡34)=34\begin{aligned} & \tan \left\{2 \tan ^{-1}\left(\frac{1}{3}\right)\right\} \\ & \because \quad 2 \tan ^{-1} x=\tan ^{-1} \frac{2 x}{1-x^2} \\ & \therefore \quad 2 \tan ^{-1}\left(\frac{1}{3}\right)=\tan ^{-1} \frac{2 / 3}{1-1 / 9} \\ & =\tan ^{-1}\left(\frac{3}{4}\right) \\ & \therefore \quad \tan \left\{2 \tan ^{-1}\left(\frac{1}{3}\right)\right\}=\tan \left(\tan \frac{3}{4}\right) \\ & =\frac{3}{4}\end{aligned}
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