Q54 / 150 · MHT CET 2022 (07 Aug Shift 2) · Mathematics

$2 \tan ^{-1}\left(\frac{1}{3}\right)+\cos ^{-1}\left(\frac{3}{5}\right)=$

  • A
    $\frac{\pi}{4}$
  • B
    $0$
  • C
    $\tan ^{-1}\left(\frac{5}{4}\right)$
  • Correct answer
    D
    $\frac{\pi}{2}$

Solution

$\begin{aligned} & 2 \tan ^{-1}\left(\frac{1}{3}\right)+\cos ^{-1}\left(\frac{3}{5}\right)=\tan ^{-1} \frac{2 \times \frac{1}{3}}{1-\left(\frac{1}{3}\right)^2}+\cos ^{-1}\left(\frac{3}{5}\right) \\ & =\tan ^{-1}\left(\frac{\frac{2}{3}}{\frac{8}{9}}\right)+\cos ^{-1}\left(\frac{3}{5}\right) \\ & =\tan ^{-1}\left(\frac{3}{4}\right)+\cot ^{-1}\left(\frac{3}{4}\right) \\ & =\frac{\pi}{2}\end{aligned}$

Hence, the correct answer is option: 'D'.

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