If 2tan−1⁡x+sin−1⁡2x1+x2 is independent of x, then

We know that          sin−1⁡2x1+x2=2tan−1⁡x,     if −1≤x≤1π−2tan−1⁡x,     if x≥1−π−2tan−1⁡x,     if x≤−1∴ 2tan−1⁡x+sin−1⁡2x1+x2=4tan−1⁡x,     if −1≤x≤1π,     if x≥1−π,     if x≤−1Thus, 2tan−1⁡x+sin−1⁡2x1+x2 is independent of x, if       x∈[1,∞) or, x∈(−∞,−1].