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

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

Minimum value of R.H.S is 1

and maximum value of  L.H.S is 1.

 so equality holds only when both sides are simultaneously equal and equal to 1.

$\begin{array}{l}\Rightarrow \cos x=1and\sin x=0\\ \text{⇒\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\cos x=1\Rightarrow x=2n\pi \\ and\sin x=0\Rightarrow x=n\pi \\ So,solutionset\text{\hspace{0.17em} is \hspace{0.17em}}x=2n\pi , n\in I\end{array}$