$\text{ The solution set of the equation }{\cos }^{5}x=1+{\sin }^{4}x\text{ is }$

${\cos }^{5}x=1+{\sin }^{4}x$

Minimum value of RHS is 1 and maximum value of LHS is 1. So equality holds only when both sides are simultaneously 1

$\text{ i.e. }\cos x=1\text{ and }\sin x=0$

$\Rightarrow x=2n\pi \text{ and }x=n\pi$

$\text{ So, solution set, }x=2n\pi$