If $\sin ^{-1}\left(\frac{x}{5}\right)+\operatorname{cosec}^{-1}\left(\frac{5}{4}\right)=\frac{\pi}{2}$,…

If $\sin ^{-1}\left(\frac{x}{5}\right)+\operatorname{cosec}^{-1}\left(\frac{5}{4}\right)=\frac{\pi}{2}$, then the value of $x$ is
  1. 4
  2. 1
  3. 5
  4. 3

Solution

$\begin{aligned} & \sin ^{-1}\left(\frac{x}{5}\right)+\operatorname{cosec}^{-1}\left(\frac{5}{4}\right)=\frac{\pi}{2} \\ & \Rightarrow \sin ^{-1}\left(\frac{x}{5}\right)+\sin ^{-1}\left(\frac{4}{5}\right)=\frac{\pi}{2} \\ & \Rightarrow \sin ^{-1}\left(\frac{x}{5}\right)=\frac{\pi}{2}-\sin ^{-1}\left(\frac{4}{5}\right) \\ & \Rightarrow \sin ^{-1}\left(\frac{x}{5}\right)=\cos ^{-1}\left(\frac{4}{5}\right) \quad \cdots\left[\because \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}\right] \\ & \Rightarrow \sin ^{-1}\left(\frac{x}{5}\right)=\sin ^{-1}\left(\frac{3}{5}\right) \cdots\left[\cos ^{-1} x=\sin ^{-1} \sqrt{1-x^2}\right] \\ & \Rightarrow x=3\end{aligned}$

Asked in: MHT CET 2024 (16 May Shift 1)

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