The greatest and least values of (sin−1x)3+(cos−1x)3 are
π332,7π38
=(sin−1x+cos−1x)3−3sin−1xcos−1x(sin−1x+cos−1x)=π38−3(sin−1xcos−1x)π2=π38−3π2sin−1x(π2−sin−1x)⇒π38+3π2[(sin−1x)2−π2sin−1x]π38+3π2[(sin−1x−π4)2]−3π332=π332+3π2(sin−1x−π4)2
So, the least value is π332.
(sin−1x−π4)2≤(3π4)2
Because when x=−1 sin−1x−π4=−π2−π4=−3π4
Greatest value is π332+9π216×3π2=7π38