Q108 / 180 · COMEDK 2013 · Mathematics

If $\sin ^{-1} \frac{x}{5}+\operatorname{cosec}^{-1} \frac{5}{4}=\frac{\pi}{2}$, then $x$ is equal to

  • A
    $1$
  • B
    $4$
  • Correct answer
    C
    $3$
  • D
    $5$

Solution

We have, $\sin ^{-1} \frac{x}{5}+\operatorname{cosec}^{-1} \frac{5}{4}=\frac{\pi}{2}$ $\Rightarrow \quad \sin ^{-1} \frac{x}{5}+\sin ^{-1} \frac{4}{5}=\frac{\pi}{2}$ $\Rightarrow \quad \sin ^{-1} \frac{x}{5}+\cos ^{-1} \frac{3}{5}=\frac{\pi}{2}$ $\Rightarrow \quad \sin ^{-1} \frac{x}{5}=\frac{\pi}{2}-\cos ^{-1} \frac{3}{5}$ $\Rightarrow \quad \sin ^{-1} \frac{x}{5}=\sin ^{-1} \frac{3}{5}$ $\Rightarrow \quad \frac{x}{5}=\frac{3}{5}$ $\Rightarrow \quad x=3$

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

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