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If $\sin ^{-1}\left(\frac{3}{x}\right)+\sin ^{-1}\left(\frac{4}{x}\right)=\frac{\pi}{2}$, then $x$ is equal to
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Verified Answer
The correct answer is:
$5$
Given that,
$\begin{aligned}
& \sin ^{-1}\left(\frac{3}{x}\right)+\sin ^{-1}\left(\frac{4}{x}\right)=\frac{\pi}{2} \\
& \therefore \sin ^{-1}\left(\frac{3}{x}\right)=\frac{\pi}{2}-\sin ^{-1}\left(\frac{4}{x}\right) \\
& \Rightarrow \sin ^{-1}\left(\frac{3}{x}\right)=\cos ^{-1}\left(\frac{4}{x}\right) \\
& \Rightarrow \sin ^{-1}\left(\frac{3}{x}\right)=\sin ^{-1}\left(\frac{\sqrt{x^2-16}}{x}\right) \\
& \Rightarrow \frac{3}{x}=\frac{\sqrt{x^2-16}}{x} \\
& \Rightarrow 9=x^2-16 \Rightarrow x^2=25 \\
& \Rightarrow x= \pm 5 \\
& \Rightarrow x=5
\end{aligned}$
($\because-5$ is not satisfied the given equation)
$\begin{aligned}
& \sin ^{-1}\left(\frac{3}{x}\right)+\sin ^{-1}\left(\frac{4}{x}\right)=\frac{\pi}{2} \\
& \therefore \sin ^{-1}\left(\frac{3}{x}\right)=\frac{\pi}{2}-\sin ^{-1}\left(\frac{4}{x}\right) \\
& \Rightarrow \sin ^{-1}\left(\frac{3}{x}\right)=\cos ^{-1}\left(\frac{4}{x}\right) \\
& \Rightarrow \sin ^{-1}\left(\frac{3}{x}\right)=\sin ^{-1}\left(\frac{\sqrt{x^2-16}}{x}\right) \\
& \Rightarrow \frac{3}{x}=\frac{\sqrt{x^2-16}}{x} \\
& \Rightarrow 9=x^2-16 \Rightarrow x^2=25 \\
& \Rightarrow x= \pm 5 \\
& \Rightarrow x=5
\end{aligned}$
($\because-5$ is not satisfied the given equation)
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