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Question:

$\sin \left(\sin ^{-1} \frac{1}{5}+\cos ^{-1} x\right)=1$

Solution:

$\sin \left(\sin ^{-1} \frac{1}{5}+\cos ^{-1} x\right)=1$

$\Rightarrow \sin ^{-1} \frac{1}{5}+\cos ^{-1} x=\sin ^{-1} 1$

$\Rightarrow \sin ^{-1} \frac{1}{5}+\cos ^{-1} x=\frac{\pi}{2}$

$\Rightarrow \sin ^{-1} \frac{1}{5}=\frac{\pi}{2}-\cos ^{-1} x$

$\Rightarrow \sin ^{-1} \frac{1}{5}=\sin ^{-1} x$                       $\left[\because \sin ^{-1} x=\frac{\pi}{2}-\cos ^{-1} x\right]$

$\Rightarrow x=\frac{1}{5}$

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