Prove that the equation, \( \left(\sin ^{-1} x\right)^{3}+\left(\cos ^{-1} x\right)^{3}=\alpha \...

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Prove that the equation, \( \left(\sin ^{-1} x\right)^{3}+\left(\cos ^{-1} x\right)^{3}=\alpha \pi^{3} \) has no roots for \( \alpha\frac{1}{32} \) and \( \alpha\frac{7}{8} \).
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