The greatest and least values of sin−1⁡x3+cos−1⁡x3 are

We have,  sin−1⁡x3+cos−1⁡x3   =sin−1⁡x+cos−1⁡x3−3sin−1⁡xcos−1⁡xsin−1⁡x+cos−1⁡x    =π38−3sin−1⁡xcos−1⁡xπ2=π38−3π2π2−sin−1⁡xsin−1⁡x   =π38−3π24sin1⁡x+3π2sin−1⁡x2  =π38+3π2sin−1⁡x2−π2sin−1⁡x=π38+3π2sin−1⁡x−π42−3π332=π332+3π2sin−1⁡x−π42So, the least value of sin−1⁡x3+cos−1⁡x3 is π332      sin−1⁡x−π42≤3π42∴   Greatest value =π332+9π216×3π2=7π38