$\text{ If }{\sin }^x\alpha +{\cos }^x\alpha \ge 1, 0<\alpha <\frac{\pi }{2},\text{ then }$

[SolutionStep1]:

${\sin }^x\alpha +{\cos }^x\alpha \ge 1,0<\alpha <\frac{\text{π}}{\text{2}}$

$\text{ Since equality holds for }x=2$

$\text{ If }x<2,\text{ both }\cos \alpha \text{ and }\sin \alpha \text{ increase (being + ve fraction). }$

${\cos }^x\alpha +{\sin }^x\alpha >1\text{ if }x<2$

$\text{ Thus }x\le 2\text{ i.e. }x\in (-\infty ,2]$