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Question:

If $\cos ^{-1} x+\cos ^{-1} y=\frac{\pi}{4}$, find the value of $\sin ^{-1} x+\sin ^{-1} y$

Solution:

$\cos ^{-1} x+\cos ^{-1} y=\frac{\pi}{4}$

$\Rightarrow \frac{\pi}{2}-\sin ^{-1} x+\frac{\pi}{2}-\sin ^{-1} y=\frac{\pi}{4} \quad\left[\because \cos ^{-1} x=\frac{\pi}{2}-\sin ^{-1} x\right]$

$\Rightarrow \pi-\left(\sin ^{-1} x+\sin ^{-1} y\right)=\frac{\pi}{4}$

$\Rightarrow \sin ^{-1} x+\sin ^{-1} y=\frac{3 \pi}{4}$

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