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If $\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}$, then $\cos ^{-1} x+\cos ^{-1} y=$
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$\frac{\pi}{3}$
$\begin{aligned} & \sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3} \quad \Rightarrow \frac{\pi}{2}-\cos ^{-1} x+\frac{\pi}{2}-\cos ^{-1} y=\frac{2 \pi}{3} \\ & \Rightarrow \cos ^{-1} x+\cos ^{-1} y=\pi-\frac{2 \pi}{3}=\frac{\pi}{3}\end{aligned}$
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