Number of values of $x\in [0,5\pi ]$ and satisfying ${\sin }^{2}x={\cos }^{2}x$ is 

$\begin{array}{l}{\tan }^{2}x=1\Rightarrow \tan x=\pm 1\\ \tan x=1\Rightarrow x=\frac{\pi }{4},\frac{5\pi }{4},\frac{9\pi }{4},\frac{13\pi }{4},\frac{17\pi }{4}\end{array}$

and $\tan x=-1\Rightarrow x=\frac{3\pi }{4},\frac{7\pi }{4},\frac{11\pi }{4},\frac{15\pi }{4},\frac{19\pi }{4}$

Thus, there are 10 values of $x.$