Q68 / 180 · COMEDK 2015 · Mathematics

$\tan \left(\cos ^{-1}\left(\frac{1}{5 \sqrt{2}}\right)-\sin ^{-1}\left(\frac{4}{\sqrt{17}}\right)\right)$ is equal to

  • A
    $\frac{\sqrt{29}}{3}$
  • B
    $\frac{29}{3}$
  • C
    $\frac{\sqrt{3}}{29}$
  • Correct answer
    D
    $\frac{3}{29}$

Solution

$\tan \left(\cos ^{-1}\left(\frac{1}{5 \sqrt{2}}\right)-\sin ^{-1}\left(\frac{4}{\sqrt{17}}\right)\right)$ $=\tan \left(\frac{\pi}{2}-\sin ^{-1}\left(\frac{1}{5 \sqrt{2}}\right)-\sin ^{-1}\left(\frac{4}{\sqrt{17}}\right)\right)$ $\quad \quad \left[\because \cos ^{-1} x=\frac{\pi}{2}-\sin ^{-1} x\right]$ $=\tan \left(\frac{\pi}{2}-\left(\sin ^{-1} \frac{4}{\sqrt{17}}+\sin ^{-1} \frac{1}{5 \sqrt{2}}\right)\right)$ $=\tan \left(\frac{\pi}{2}-\sin ^{-1}\left(\frac{4}{\sqrt{17}} \cdot \sqrt{1-\frac{1}{5}}+\frac{\pi}{2}-\sin ^{-1} x\right]\right.$ $=\tan \left(\frac{\pi}{2}-\sin ^{-1}\left(\frac{4}{\sqrt{17}} \cdot \frac{1}{5 \sqrt{2}}+\frac{1}{5 \sqrt{2}} \cdot \frac{1}{\sqrt{17}}\right)\right)$ $\left.=\cot \left(\sin ^{-1} \frac{29}{5 \sqrt{2} \cdot \sqrt{17}}\right)\right)$ $\left.=\cot \left(\cot ^{-1} \frac{3}{29}\right)=\frac{3}{29}\right)$

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

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